User:Jemimah adams

CIVIC EDUCATION Class: Basic 8 Subject: Civic Education. Topic: The Rule of law. - Definition of the rule of law. The rule of law is the supremacy of law over everybody in a political system. It is also seen as the total predominance or supremacy of ordinary law of the land over all the citizens, no matter his or her status. Professor A.V Dicey from United Kingdom propounded the rule of law in 1885. – The Principles Of The Rule of Law. The following are the principles of the rule of law. 1. Principle of equality 2. Principle of impartiality 3. Principle of individual rights. 4. Supremacy of law. 5. Fair hearing. etc.

Date : 21/07/20 Class: Basic 8 Subject : Civic Education Topic: The Rule Of Law (cont) - Benefit/ Importance Of The Rule Of Law The following are the benefits or importance of the rule of law. i.	it promotes national unity ii. . it brings economic, social and political development iii. the rule of law protects the equality of all citizens before the law. iv. it gives the country great image abroad. v.	 the rule of law serves as a legal guide for citizens in relating to themselves.

- Agencies Responsible for For Protecting The Rule Of Law And Human Rights The agencies are as follow; i. the Judiciary- court. ii. human right commission. iii. the police.

- Definition of Leadership This is the ability to guide or influence people of a particular organisation,community, society or country to archive desired goals. - Qualities of a Good Leader. A good leader must posses the following qualities. i.   discipline. vi. dedication ii. honesty                        vii.responsibility. iii. ability to co-ordinate iv. punctuality v.  integrity

- consequences of of Bad Leadership. The following are consequences of bad leadership on followers. i. civil unrest in the polity. ii. economic hardship. iii. Ioss of authority when follower withdraws its support. iv. corruption and indiscipline in the society. v. lack of patriotism and nationalism on     the part of the followers. etc.

Date: 8/09/2020 Class: Basic 8 Subject: Civic Education. Topic:  Poverty ( cont )

- Ways Of Alleviating Poverty. The following are some of the ways or strategies of solving the problems of poverty. i. free qualitative education: government should ensure free universal basic education to enable children from both poor and rich home to receive basic education. ii. establishment of skill acquisition center: government should create more skill or vocational centers for less privilege. iii. good pay for workers: restructuring better salary scale for workers of low grade   levels. iv. good saving habits. learn to save for future. v. establishment of job - generating industries and agencies

ANGLES

Not to be confused with Angel. This article is about angles in geometry. For other uses, see Angle (disambiguation).

An angle formed by two rays emanating from a vertex. In plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.[1] Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane. Angles are also formed by the intersection of two planes in Euclidean and other spaces. These are called dihedral angles. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements also hold in space. For example, the spherical angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles.

Angle is also used to designate the measure of an angle or of a rotation. This measure is the ratio of the length of a circular arc to its radius. In the case of a geometric angle, the arc is centered at the vertex and delimited by the sides. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation.

The word angle comes from the Latin word angulus, meaning "corner"; cognate words are the Greek ἀγκύλος (ankylοs), meaning "crooked, curved," and the English word "ankle". Both are connected with the Proto-Indo-European root *ank-, meaning "to bend" or "bow".[2]

Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. According to Proclus, an angle must be either a quality or a quantity, or a relationship. The first concept was used by Eudemus, who regarded an angle as a deviation from a straight line; the second by Carpus of Antioch, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third concept, although his definitions of right, acute, and obtuse angles are certainly quantitative.[3]

Contents 1	Identifying angles 2	Types of angles 2.1	Individual angles 2.2	Equivalence angle pairs 2.3	Vertical and adjacent angle pairs 2.4	Combining angle pairs 2.5	Polygon-related angles 2.6	Plane-related angles 3	Measuring angles 3.1	Angle addition postulate 3.2	Units 3.3	Positive and negative angles 3.4	Alternative ways of measuring the size of an angle 3.5	Astronomical approximations 4	Angles between curves 5	Bisecting and trisecting angles 6	Dot product and generalisations 6.1	Inner product 6.2	Angles between subspaces 6.3	Angles in Riemannian geometry 6.4	Hyperbolic angle 7	Angles in geography and astronomy 8	See also 9	Notes 10	References 11	Bibliography 12	External links Identifying angles In mathematical expressions, it is common to use Greek letters (α, β, γ, θ, φ, . . . ) as variables denoting the size of some angle[4] (to avoid confusion with its other meaning, the symbol π is typically not used for this purpose). Lower case Roman letters (a, b, c, . . . ) are also used, as are upper case Roman letters in the context of polygons. See the figures in this article for examples.

In geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB and AC (i.e. the lines from point A to point B and point A to point C) is denoted ∠BAC (in Unicode U+2220 ∠ ANGLE) or {\displaystyle {\widehat {\rm {BAC}}}}{\displaystyle {\widehat {\rm {BAC}}}}. Where there is no risk of confusion, the angle may sometimes be referred to simply by its vertex (in this case "angle A").

Potentially, an angle denoted as, say, ∠BAC, might refer to any of four angles: the clockwise angle from B to C, the anticlockwise angle from B to C, the clockwise angle from C to B, or the anticlockwise angle from C to B, where the direction in which the angle is measured determines its sign (see Positive and negative angles). However, in many geometrical situations, it is obvious from context that the positive angle less than or equal to 180 degrees is meant, in which case no ambiguity arises. Otherwise, a convention may be adopted so that ∠BAC always refers to the anticlockwise (positive) angle from B to C, and ∠CAB the anticlockwise (positive) angle from C to B.

Types of angles "Oblique angle" redirects here. For the cinematographic technique, see Dutch angle. Individual angles There is some common terminology for angles, whose measure is always non-negative (see #Positive and negative angles):[5][6]

An angle equal to 0° or not turned is called a zero angle. Angles smaller than a right angle (less than 90°) are called acute angles ("acute" meaning "sharp"). An angle equal to 1 / 4 turn (90° or π / 2 radians) is called a right angle. Two lines that form a right angle are said to be normal, orthogonal, or perpendicular. Angles larger than a right angle and smaller than a straight angle (between 90° and 180°) are called obtuse angles ("obtuse" meaning "blunt"). An angle equal to 1 / 2 turn (180° or π radians) is called a straight angle. Angles larger than a straight angle but less than 1 turn (between 180° and 360°) are called reflex angles. An angle equal to 1 turn (360° or 2π radians) is called a full angle, complete angle, round angle or a perigon. Angles that are not right angles or a multiple of a right angle are called oblique angles. The names, intervals, and measured units are shown in the table below:

Right angle

Acute (a), obtuse (b), and straight (c) angles. The acute and obtuse angles are also known as oblique angles.

Reflex angle Name 	zero	acute	right angle	obtuse	straight	reflex	perigon Units	Interval Turns 	0	(0,  1 / 4 ) 1 / 4 ( 1 / 4, 1 / 2 )	1 / 2 ( 1 / 2 ,  1)	1 Radians	0	(0, 1 / 2 π) 1 / 2 π	( 1 / 2 π, π)	π	(π, 2π)	2π Degrees 	0°	(0, 90)°	90°	(90, 180)°	180°	(180, 360)°	360° Gons 	0g	(0, 100)g	100g	(100, 200)g	200g	(200, 400)g	400g Equivalence angle pairs Angles that have the same measure (i.e. the same magnitude) are said to be equal or congruent. An angle is defined by its measure and is not dependent upon the lengths of the sides of the angle (e.g. all right angles are equal in measure). Two angles which share terminal sides, but differ in size by an integer multiple of a turn, are called coterminal angles. A reference angle is the acute version of any angle determined by repeatedly subtracting or adding straight angle ( 1 / 2 turn, 180°, or π radians), to the results as necessary, until the magnitude of result is an acute angle, a value between 0 and 1 / 4 turn, 90°, or π / 2 radians. For example, an angle of 30 degrees has a reference angle of 30 degrees, and an angle of 150 degrees also has a reference angle of 30 degrees (180–150). An angle of 750 degrees has a reference angle of 30 degrees (750–720).[7] Vertical and adjacent angle pairs

Angles A and B are a pair of vertical angles; angles C and D are a pair of vertical angles. When two straight lines intersect at a point, four angles are formed. Pairwise these angles are named according to their location relative to each other.

A pair of angles opposite each other, formed by two intersecting straight lines that form an "X"-like shape, are called vertical angles or opposite angles or vertically opposite angles. They are abbreviated as vert. opp. ∠s.[8] The equality of vertically opposite angles is called the vertical angle theorem. Eudemus of Rhodes attributed the proof to Thales of Miletus.[9][10] The proposition showed that since both of a pair of vertical angles are supplementary to both of the adjacent angles, the vertical angles are equal in measure. According to a historical note,[10] when Thales visited Egypt, he observed that whenever the Egyptians drew two intersecting lines, they would measure the vertical angles to make sure that they were equal. Thales concluded that one could prove that all vertical angles are equal, if one accepted some general notions such as: All straight angles are equal. Equals added to equals are equal. Equals subtracted from equals are equal. When two adjacent angles form a straight line, they are supplementary. Therefore, if we assume that the measure of angle A equals x, then the measure of angle C would be 180 − x. Similarly, the measure of angle D would be 180 − x. Both angle C and angle D have measures equal to 180 − x and are congruent. Since angle B is supplementary to both angles C and D, either of these angle measures may be used to determine the measure of Angle B. Using the measure of either angle C or angle D, we find the measure of angle B to be 180 − (180 − x) = 180 − 180 + x = x. Therefore, both angle A and angle B have measures equal to x and are equal in measure.

Angles A and B are adjacent. Adjacent angles, often abbreviated as adj. ∠s, are angles that share a common vertex and edge but do not share any interior points. In other words, they are angles that are side by side, or adjacent, sharing an "arm". Adjacent angles which sum to a right angle, straight angle or full angle are special, and are respectively called complementary, supplementary and explementary angles (see "Combine angle pairs" below). A transversal is a line that intersects a pair of (often parallel) lines, and is associated with alternate interior angles, corresponding angles, interior angles, and exterior angles.[11]

Combining angle pairs There are three special angle pairs which involve the summation of angles:

The complementary angles a and b (b is the complement of a, and a is the complement of b). Complementary angles are angle pairs whose measures sum to one right angle ( 1 / 4 turn, 90°, or π / 2 radians).[12] If the two complementary angles are adjacent, their non-shared sides form a right angle. In Euclidean geometry, the two acute angles in a right triangle are complementary, because the sum of internal angles of a triangle is 180 degrees, and the right angle itself accounts for 90 degrees. The adjective complementary is from Latin complementum, associated with the verb complere, "to fill up". An acute angle is "filled up" by its complement to form a right angle. The difference between an angle and a right angle is termed the complement of the angle.[13] If angles A and B are complementary, the following relationships hold: {\displaystyle {\begin{aligned}&\sin ^{2}A+\sin ^{2}B=1&&\cos ^{2}A+\cos ^{2}B=1\\[3pt]&\tan A=\cot B&&\sec A=\csc B\end{aligned}}}{\displaystyle {\begin{aligned}&\sin ^{2}A+\sin ^{2}B=1&&\cos ^{2}A+\cos ^{2}B=1\\[3pt]&\tan A=\cot B&&\sec A=\csc B\end{aligned}}} (The tangent of an angle equals the cotangent of its complement and its secant equals the cosecant of its complement.) The prefix "co-" in the names of some trigonometric ratios refers to the word "complementary".

The angles a and b are supplementary angles. Two angles that sum to a straight angle ( 1 / 2 turn, 180°, or π radians) are called supplementary angles.[14] If the two supplementary angles are adjacent (i.e. have a common vertex and share just one side), their non-shared sides form a straight line. Such angles are called a linear pair of angles.[15] However, supplementary angles do not have to be on the same line, and can be separated in space. For example, adjacent angles of a parallelogram are supplementary, and opposite angles of a cyclic quadrilateral (one whose vertices all fall on a single circle) are supplementary. If a point P is exterior to a circle with center O, and if the tangent lines from P touch the circle at points T and Q, then ∠TPQ and ∠TOQ are supplementary. The sines of supplementary angles are equal. Their cosines and tangents (unless undefined) are equal in magnitude but have opposite signs. In Euclidean geometry, any sum of two angles in a triangle is supplementary to the third, because the sum of internal angles of a triangle is a straight angle.

Sum of two explementary angles is a complete angle. Two angles that sum to a complete angle (1 turn, 360°, or 2π radians) are called explementary angles or conjugate angles. The difference between an angle and a complete angle is termed the explement of the angle or conjugate of an angle. Polygon-related angles

Internal and external angles. An angle that is part of a simple polygon is called an interior angle if it lies on the inside of that simple polygon. A simple concave polygon has at least one interior angle that is a reflex angle. In Euclidean geometry, the measures of the interior angles of a triangle add up to π radians, 180°, or 1 / 2 turn; the measures of the interior angles of a simple convex quadrilateral add up to 2π radians, 360°, or 1 turn. In general, the measures of the interior angles of a simple convex polygon with n sides add up to (n − 2)π radians, or 180(n − 2) degrees, (2n − 4) right angles, or ( n / 2 − 1) turn. The supplement of an interior angle is called an exterior angle, that is, an interior angle and an exterior angle form a linear pair of angles. There are two exterior angles at each vertex of the polygon, each determined by extending one of the two sides of the polygon that meet at the vertex; these two angles are vertical angles and hence are equal. An exterior angle measures the amount of rotation one has to make at a vertex to trace out the polygon.[16] If the corresponding interior angle is a reflex angle, the exterior angle should be considered negative. Even in a non-simple polygon it may be possible to define the exterior angle, but one will have to pick an orientation of the plane (or surface) to decide the sign of the exterior angle measure. In Euclidean geometry, the sum of the exterior angles of a simple convex polygon will be one full turn (360°). The exterior angle here could be called a supplementary exterior angle. Exterior angles are commonly used in Logo Turtle programs when drawing regular polygons. In a triangle, the bisectors of two exterior angles and the bisector of the other interior angle are concurrent (meet at a single point).[17]:p. 149 In a triangle, three intersection points, each of an external angle bisector with the opposite extended side, are collinear.[17]:p. 149 In a triangle, three intersection points, two of them between an interior angle bisector and the opposite side, and the third between the other exterior angle bisector and the opposite side extended, are collinear.[17]:p. 149 Some authors use the name exterior angle of a simple polygon to simply mean the explement exterior angle (not supplement!) of the interior angle.[18] This conflicts with the above usage. Plane-related angles The angle between two planes (such as two adjacent faces of a polyhedron) is called a dihedral angle.[13] It may be defined as the acute angle between two lines normal to the planes. The angle between a plane and an intersecting straight line is equal to ninety degrees minus the angle between the intersecting line and the line that goes through the point of intersection and is normal to the plane. Measuring angles

It has been suggested that Angular unit be merged into this article. (Discuss) Proposed since May 2020. The size of a geometric angle is usually characterized by the magnitude of the smallest rotation that maps one of the rays into the other. Angles that have the same size are said to be equal or congruent or equal in measure.

In some contexts, such as identifying a point on a circle or describing the orientation of an object in two dimensions relative to a reference orientation, angles that differ by an exact multiple of a full turn are effectively equivalent. In other contexts, such as identifying a point on a spiral curve or describing the cumulative rotation of an object in two dimensions relative to a reference orientation, angles that differ by a non-zero multiple of a full turn are not equivalent.

The measure of angle θ (in radians) is the quotient of s and r. In order to measure an angle θ, a circular arc centered at the vertex of the angle is drawn, e.g. with a pair of compasses. The ratio of the length s of the arc by the radius r of the circle is the measure of the angle in radians.

The measure of the angle in another angular unit is then obtained by multiplying its measure in radians by the scaling factor k / 2π , where k is the measure of a complete turn in the chosen unit (for example 360 for degrees or 400 for gradians):

{\displaystyle \theta =k{\frac {s}{2\pi r}}.}\theta =k{\frac {s}{2\pi r}}. The value of θ thus defined is independent of the size of the circle: if the length of the radius is changed then the arc length changes in the same proportion, so the ratio s/r is unaltered. (Proof. The formula above can be rewritten as k = θr / s . One turn, for which θ = n units, corresponds to an arc equal in length to the circle's circumference, which is 2πr, so s = 2πr. Substituting n for θ and 2πr for s in the formula, results in k = nr / 2πr = n / 2π .) [nb 1]

Angle addition postulate The angle addition postulate states that if B is in the interior of angle AOC, then

{\displaystyle m\angle AOC=m\angle AOB+m\angle BOC}m\angle AOC=m\angle AOB+m\angle BOC The measure of the angle AOC is the sum of the measure of angle AOB and the measure of angle BOC. In this postulate it does not matter in which unit the angle is measured as long as each angle is measured in the same unit.

Units See also: Angular unit Units used to represent angles are listed below in descending magnitude order. Of these units, the degree and the radian are by far the most commonly used. Angles expressed in radians are dimensionless for the purposes of dimensional analysis.

Most units of angular measurement are defined such that one turn (i.e. one full circle) is equal to n units, for some whole number n. The two exceptions are the radian and the diameter part.

Turn (n = 1) The turn, also cycle, full circle, revolution, and rotation, is complete circular movement or measure (as to return to the same point) with circle or ellipse. A turn is abbreviated τ, cyc, rev, or rot depending on the application, but in the acronym rpm (revolutions per minute), just r is used. A turn of n units is obtained by setting k = 1 / 2π in the formula above. The equivalence of 1 turn is 360°, 2π rad, 400 grad, and 4 right angles. The symbol τ can also be used as a mathematical constant to represent 2π radians. Used in this way (k = τ / 2π ) allows for radians to be expressed as a fraction of a turn. For example, half a turn is τ / 2 = π. Quadrant (n = 4) The quadrant is 1 / 4 of a turn, i.e. a right angle. It is the unit used in Euclid's Elements. 1 quad. = 90° = π / 2 rad = 1 / 4 turn = 100 grad. In German the symbol ∟ has been used to denote a quadrant. Sextant (n = 6) The sextant (angle of the equilateral triangle) is 1 / 6 of a turn. It was the unit used by the Babylonians,[20][21] and is especially easy to construct with ruler and compasses. The degree, minute of arc and second of arc are sexagesimal subunits of the Babylonian unit. 1 Babylonian unit = 60° = π/3 rad ≈ 1.047197551 rad.

θ = s/r rad = 1 rad. Radian (n = 2π =  6.283 . . . ) The radian is the angle subtended by an arc of a circle that has the same length as the circle's radius. The case of radian for the formula given earlier, a radian of n = 2π units is obtained by setting k = 2π / 2π = 1. One turn is 2π radians, and one radian is 180 / π degrees, or about 57.2958 degrees. The radian is abbreviated rad, though this symbol is often omitted in mathematical texts, where radians are assumed unless specified otherwise. When radians are used angles are considered as dimensionless. The radian is used in virtually all mathematical work beyond simple practical geometry, due, for example, to the pleasing and "natural" properties that the trigonometric functions display when their arguments are in radians. The radian is the (derived) unit of angular measurement in the SI system. Clock position (n = 12) A clock position is the relative direction of an object described using the analogy of a 12-hour clock. One imagines a clock face lying either upright or flat in front of oneself, and identifies the twelve hour markings with the directions in which they point. Hour angle (n = 24) The astronomical hour angle is 1 / 24 of a turn. As this system is amenable to measuring objects that cycle once per day (such as the relative position of stars), the sexagesimal subunits are called minute of time and second of time. These are distinct from, and 15 times larger than, minutes and seconds of arc. 1 hour = 15° = π / 12 rad = 1 / 6 quad. = 1 / 24 turn = 16+ 2 / 3 grad. (Compass) point or wind (n = 32) The point, used in navigation, is 1 / 32 of a turn. 1 point = 1 / 8 of a right angle = 11.25° = 12.5 grad. Each point is subdivided in four quarter-points so that 1 turn equals 128 quarter-points. Hexacontade (n = 60) The hexacontade is a unit of 6° that Eratosthenes used, so that a whole turn was divided into 60 units. Pechus (n = 144–180) The pechus was a Babylonian unit equal to about 2° or 2+ 1 / 2 °. Binary degree (n = 256) The binary degree, also known as the binary radian (or brad), is 1 / 256 of a turn.[22] The binary degree is used in computing so that an angle can be efficiently represented in a single byte (albeit to limited precision). Other measures of angle used in computing may be based on dividing one whole turn into 2n equal parts for other values of n.[23] Degree (n = 360) The degree, denoted by a small superscript circle (°), is 1/360 of a turn, so one turn is 360°. The case of degrees for the formula given earlier, a degree of n = 360° units is obtained by setting k = 360° / 2π . One advantage of this old sexagesimal subunit is that many angles common in simple geometry are measured as a whole number of degrees. Fractions of a degree may be written in normal decimal notation (e.g. 3.5° for three and a half degrees), but the "minute" and "second" sexagesimal subunits of the "degree-minute-second" system are also in use, especially for geographical coordinates and in astronomy and ballistics. Diameter part (n = 376.99 . . . ) The diameter part (occasionally used in Islamic mathematics) is 1 / 60 radian. One "diameter part" is approximately 0.95493°. There are about 376.991 diameter parts per turn. Grad (n = 400) The grad, also called grade, gradian, or gon, is 1 / 400 of a turn, so a right angle is 100 grads.[4] It is a decimal subunit of the quadrant. A kilometre was historically defined as a centi-grad of arc along a great circle of the Earth, so the kilometer is the decimal analog to the sexagesimal nautical mile. The grad is used mostly in triangulation. Milliradian The milliradian (mil or mrad) is defined as a thousandth of a radian, which means that a rotation of one turn consists of 2000π mil (or approximately 6283.185... mil), and almost all scope sights for firearms are calibrated to this definition. In addition there are three other derived definitions used for artillery and navigation which are approximately equal to a milliradian. Under these three other definitions one turn makes up for exactly 6000, 6300 or 6400 mils, which equals spanning the range from 0.05625 to 0.06 degrees (3.375 to 3.6 minutes). In comparison, the true milliradian is approximately 0.05729578... degrees (3.43775... minutes). One "NATO mil" is defined as 1 / 6400 of a circle. Just like with the true milliradian, each of the other definitions exploits the mil's handby property of subtensions, i.e. that the value of one milliradian approximately equals the angle subtended by a width of 1 meter as seen from 1 km away ( 2π / 6400 = 0.0009817... ≈ 1 / 1000 ). Minute of arc (n = 21,600) The minute of arc (or MOA, arcminute, or just minute) is 1 / 60 of a degree = 1 / 21,600 turn. It is denoted by a single prime ( ′ ). For example, 3° 30′ is equal to 3 × 60 + 30 = 210 minutes or 3 + 30 / 60 = 3.5 degrees. A mixed format with decimal fractions is also sometimes used, e.g. 3° 5.72′ = 3 + 5.72 / 60 degrees. A nautical mile was historically defined as a minute of arc along a great circle of the Earth. Second of arc (n = 1,296,000) The second of arc (or arcsecond, or just second) is 1 / 60 of a minute of arc and 1 / 3600 of a degree. It is denoted by a double prime ( ″ ). For example, 3° 7′ 30″ is equal to 3 + 7 / 60 + 30 / 3600 degrees, or 3.125 degrees. Milliarcsecond (n = 1,296,000,000) mas Microarcsecond (n = 1,296,000,000,000) µas Positive and negative angles Although the definition of the measurement of an angle does not support the concept of a negative angle, it is frequently useful to impose a convention that allows positive and negative angular values to represent orientations and/or rotations in opposite directions relative to some reference.

In a two-dimensional Cartesian coordinate system, an angle is typically defined by its two sides, with its vertex at the origin. The initial side is on the positive x-axis, while the other side or terminal side is defined by the measure from the initial side in radians, degrees, or turns. With positive angles representing rotations toward the positive y-axis and negative angles representing rotations toward the negative y-axis. When Cartesian coordinates are represented by standard position, defined by the x-axis rightward and the y-axis upward, positive rotations are anticlockwise and negative rotations are clockwise.

In many contexts, an angle of −θ is effectively equivalent to an angle of "one full turn minus θ". For example, an orientation represented as −45° is effectively equivalent to an orientation represented as 360° − 45° or 315°. Although the final position is the same, a physical rotation (movement) of −45° is not the same as a rotation of 315° (for example, the rotation of a person holding a broom resting on a dusty floor would leave visually different traces of swept regions on the floor).

In three-dimensional geometry, "clockwise" and "anticlockwise" have no absolute meaning, so the direction of positive and negative angles must be defined relative to some reference, which is typically a vector passing through the angle's vertex and perpendicular to the plane in which the rays of the angle lie.

In navigation, bearings or azimuth are measured relative to north. By convention, viewed from above, bearing angles are positive clockwise, so a bearing of 45° corresponds to a north-east orientation. Negative bearings are not used in navigation, so a north-west orientation corresponds to a bearing of 315°.

Alternative ways of measuring the size of an angle There are several alternatives to measuring the size of an angle by the angle of rotation. The grade of a slope, or gradient is equal to the tangent of the angle, or sometimes (rarely) the sine. A gradient is often expressed as a percentage. For very small values (less than 5%), the grade of a slope is approximately the measure of the angle in radians.

In rational geometry the spread between two lines is defined as the square of the sine of the angle between the lines. As the sine of an angle and the sine of its supplementary angle are the same, any angle of rotation that maps one of the lines into the other leads to the same value for the spread between the lines.

Astronomical approximations Main article: Angular diameter Astronomers measure angular separation of objects in degrees from their point of observation.

0.5° is approximately the width of the sun or moon. 1° is approximately the width of a little finger at arm's length. 10° is approximately the width of a closed fist at arm's length. 20° is approximately the width of a handspan at arm's length. These measurements clearly depend on the individual subject, and the above should be treated as rough rule of thumb approximations only.

Angles between curves

The angle between the two curves at P is defined as the angle between the tangents A and B at P. The angle between a line and a curve (mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the tangents at the point of intersection. Various names (now rarely, if ever, used) have been given to particular cases:—amphicyrtic (Gr. ἀμφί, on both sides, κυρτός, convex) or cissoidal (Gr. κισσός, ivy), biconvex; xystroidal or sistroidal (Gr. ξυστρίς, a tool for scraping), concavo-convex; amphicoelic (Gr. κοίλη, a hollow) or angulus lunularis, biconcave.[24]

Bisecting and trisecting angles Main articles: Bisection § Angle bisector, and Angle trisection The ancient Greek mathematicians knew how to bisect an angle (divide it into two angles of equal measure) using only a compass and straightedge, but could only trisect certain angles. In 1837 Pierre Wantzel showed that for most angles this construction cannot be performed.

Dot product and generalisations In the Euclidean space, the angle θ between two Euclidean vectors u and v is related to their dot product and their lengths by the formula

{\displaystyle \mathbf {u} \cdot \mathbf {v} =\cos(\theta )\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}{\displaystyle \mathbf {u} \cdot \mathbf {v} =\cos(\theta )\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.} This formula supplies an easy method to find the angle between two planes (or curved surfaces) from their normal vectors and between skew lines from their vector equations.

Inner product To define angles in an abstract real inner product space, we replace the Euclidean dot product ( · ) by the inner product {\displaystyle \langle \cdot ,\cdot \rangle }\langle \cdot ,\cdot \rangle, i.e.

{\displaystyle \langle \mathbf {u} ,\mathbf {v} \rangle =\cos(\theta )\ \left\|\mathbf {u} \right\|\ \left\|\mathbf {v} \right\|.}\langle \mathbf {u} ,\mathbf {v} \rangle =\cos(\theta )\ \left\|\mathbf {u} \right\|\ \left\|\mathbf {v} \right\|. In a complex inner product space, the expression for the cosine above may give non-real values, so it is replaced with

{\displaystyle \operatorname {Re} \left(\langle \mathbf {u} ,\mathbf {v} \rangle \right)=\cos(\theta )\ \left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}{\displaystyle \operatorname {Re} \left(\langle \mathbf {u} ,\mathbf {v} \rangle \right)=\cos(\theta )\ \left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.} or, more commonly, using the absolute value, with

{\displaystyle \left|\langle \mathbf {u} ,\mathbf {v} \rangle \right|=|\cos(\theta )|\ \left\|\mathbf {u} \right\|\ \left\|\mathbf {v} \right\|.}\left|\langle \mathbf {u} ,\mathbf {v} \rangle \right|=|\cos(\theta )|\ \left\|\mathbf {u} \right\|\ \left\|\mathbf {v} \right\|. The latter definition ignores the direction of the vectors and thus describes the angle between one-dimensional subspaces {\displaystyle \operatorname {span} (\mathbf {u} )}\operatorname {span} (\mathbf {u} ) and {\displaystyle \operatorname {span} (\mathbf {v} )}\operatorname {span} (\mathbf {v} ) spanned by the vectors {\displaystyle \mathbf {u} }\mathbf {u} and {\displaystyle \mathbf {v} }\mathbf {v}  correspondingly.

Angles between subspaces The definition of the angle between one-dimensional subspaces {\displaystyle \operatorname {span} (\mathbf {u} )}\operatorname {span} (\mathbf {u} ) and {\displaystyle \operatorname {span} (\mathbf {v} )}\operatorname {span} (\mathbf {v} ) given by

{\displaystyle \left|\langle \mathbf {u} ,\mathbf {v} \rangle \right|=|\cos(\theta )|\left\|\mathbf {u} \right\|\ \left\|\mathbf {v} \right\|}{\displaystyle \left|\langle \mathbf {u} ,\mathbf {v} \rangle \right|=|\cos(\theta )|\left\|\mathbf {u} \right\|\ \left\|\mathbf {v} \right\|} in a Hilbert space can be extended to subspaces of any finite dimensions. Given two subspaces {\displaystyle {\mathcal {U}}}{\mathcal {U}}, {\displaystyle {\mathcal {W}}}{\mathcal {W}} with {\displaystyle \dim({\mathcal {U}}):=k\leq \dim({\mathcal {W}}):=l}{\displaystyle \dim({\mathcal {U}}):=k\leq \dim({\mathcal {W}}):=l}, this leads to a definition of {\displaystyle k}k angles called canonical or principal angles between subspaces.

Angles in Riemannian geometry In Riemannian geometry, the metric tensor is used to define the angle between two tangents. Where U and V are tangent vectors and gij are the components of the metric tensor G,

{\displaystyle \cos \theta ={\frac {g_{ij}U^{i}V^{j}}{\sqrt {\left|g_{ij}U^{i}U^{j}\right|\left|g_{ij}V^{i}V^{j}\right|}}}.}{\displaystyle \cos \theta ={\frac {g_{ij}U^{i}V^{j}}{\sqrt {\left|g_{ij}U^{i}U^{j}\right|\left|g_{ij}V^{i}V^{j}\right|}}}.} Hyperbolic angle A hyperbolic angle is an argument of a hyperbolic function just as the circular angle is the argument of a circular function. The comparison can be visualized as the size of the openings of a hyperbolic sector and a circular sector since the areas of these sectors correspond to the angle magnitudes in each case. Unlike the circular angle, the hyperbolic angle is unbounded. When the circular and hyperbolic functions are viewed as infinite series in their angle argument, the circular ones are just alternating series forms of the hyperbolic functions. This weaving of the two types of angle and function was explained by Leonhard Euler in Introduction to the Analysis of the Infinite.

Angles in geography and astronomy In geography, the location of any point on the Earth can be identified using a geographic coordinate system. This system specifies the latitude and longitude of any location in terms of angles subtended at the centre of the Earth, using the equator and (usually) the Greenwich meridian as references.

In astronomy, a given point on the celestial sphere (that is, the apparent position of an astronomical object) can be identified using any of several astronomical coordinate systems, where the references vary according to the particular system. Astronomers measure the angular separation of two stars by imagining two lines through the centre of the Earth, each intersecting one of the stars. The angle between those lines can be measured, and is the angular separation between the two stars.

In both geography and astronomy, a sighting direction can be specified in terms of a vertical angle such as altitude /elevation with respect to the horizon as well as the azimuth with respect to north.

Astronomers also measure the apparent size of objects as an angular diameter. For example, the full moon has an angular diameter of approximately 0.5°, when viewed from Earth. One could say, "The Moon's diameter subtends an angle of half a degree." The small-angle formula can be used to convert such an angular measurement into a distance/size ratio.

See also Angle bisector Angular velocity Argument (complex analysis) Astrological aspect Central angle Clock angle problem Dihedral angle Exterior angle theorem Golden angle Great circle distance Inscribed angle Irrational angle Phase (waves) Protractor Solid angle for a concept of angle in three dimensions. Spherical angle Transcendent angle Trisection Zenith angle Notes This approach requires however an additional proof that the measure of the angle does not change with changing radius r, in addition to the issue of "measurement units chosen." A smoother approach is to measure the angle by the length of the corresponding unit circle arc. Here "unit" can be chosen to be dimensionless in the sense that it is the real number 1 associated with the unit segment on the real line. See Radoslav M. Dimitrić for instance.[19] References Sidorov 2001 Slocum 2007 Chisholm 1911; Heiberg 1908, pp. 177–178 "Compendium of Mathematical Symbols". Math Vault. 2020-03-01. Retrieved 2020-08-17. "Angles - Acute, Obtuse, Straight and Right". www.mathsisfun.com. Retrieved 2020-08-17. Weisstein, Eric W. "Angle". mathworld.wolfram.com. Retrieved 2020-08-17. "Mathwords: Reference Angle". www.mathwords.com. Archived from the original on 23 October 2017. Retrieved 26 April 2018. Wong & Wong 2009, pp. 161–163 Euclid. The Elements. Proposition I:13. Shute, Shirk & Porter 1960, pp. 25–27. Jacobs 1974, p. 255. "Complementary Angles". www.mathsisfun.com. Retrieved 2020-08-17. Chisholm 1911 "Supplementary Angles". www.mathsisfun.com. Retrieved 2020-08-17. Jacobs 1974, p. 97. Henderson & Taimina 2005, p. 104. Johnson, Roger A. Advanced Euclidean Geometry, Dover Publications, 2007. D. Zwillinger, ed. (1995), CRC Standard Mathematical Tables and Formulae, Boca Raton, FL: CRC Press, p. 270 as cited in Weisstein, Eric W. "Exterior Angle". MathWorld. Dimitrić, Radoslav M. (2012). "On Angles and Angle Measurements" (PDF). The Teaching of Mathematics. XV (2): 133–140. Archived (PDF) from the original on 2019-01-17. Retrieved 2019-08-06. Jeans, James Hopwood (1947). The Growth of Physical Science. CUP Archive. p. 7. Murnaghan, Francis Dominic (1946). Analytic Geometry. p. 2. "ooPIC Programmer's Guide - Chapter 15: URCP". ooPIC Manual & Technical Specifications - ooPIC Compiler Ver 6.0. Savage Innovations, LLC. 2007 [1997]. Archived from the original on 2008-06-28. Retrieved 2019-08-05. Hargreaves, Shawn. "Angles, integers, and modulo arithmetic". blogs.msdn.com. Archived from the original on 2019-06-30. Retrieved 2019-08-05. Chisholm 1911; Heiberg 1908, p. 178 Bibliography Henderson, David W.; Taimina, Daina (2005), Experiencing Geometry / Euclidean and Non-Euclidean with History (3rd ed.), Pearson Prentice Hall, p. 104, ISBN 978-0-13-143748-7 Heiberg, Johan Ludvig (1908), Heath, T. L. (ed.), Euclid, The Thirteen Books of Euclid's Elements, 1, Cambridge: Cambridge University Press. Sidorov, L. A. (2001) [1994], "Angle", Encyclopedia of Mathematics, EMS Press Jacobs, Harold R. (1974), Geometry, W. H. Freeman, pp. 97, 255, ISBN 978-0-7167-0456-0 Slocum, Jonathan (2007), Preliminary Indo-European lexicon — Pokorny PIE data, University of Texas research department: linguistics research center, retrieved 2 Feb 2010 Shute, William G.; Shirk, William W.; Porter, George F. (1960), Plane and Solid Geometry, American Book Company, pp. 25–27 Wong, Tak-wah; Wong, Ming-sim (2009), "Angles in Intersecting and Parallel Lines", New Century Mathematics, 1B (1 ed.), Hong Kong: Oxford University Press, pp. 161–163, ISBN 978-0-19-800177-5 This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911), "Angle", Encyclopædia Britannica, 2 (11th ed.), Cambridge University Press, p. 14

AGRIC

Not to be confused with Angel. This article is about angles in geometry. For other uses, see Angle (disambiguation).

An angle formed by two rays emanating from a vertex. In plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.[1] Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane. Angles are also formed by the intersection of two planes in Euclidean and other spaces. These are called dihedral angles. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements also hold in space. For example, the spherical angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles.

Angle is also used to designate the measure of an angle or of a rotation. This measure is the ratio of the length of a circular arc to its radius. In the case of a geometric angle, the arc is centered at the vertex and delimited by the sides. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation.

The word angle comes from the Latin word angulus, meaning "corner"; cognate words are the Greek ἀγκύλος (ankylοs), meaning "crooked, curved," and the English word "ankle". Both are connected with the Proto-Indo-European root *ank-, meaning "to bend" or "bow".[2]

Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. According to Proclus, an angle must be either a quality or a quantity, or a relationship. The first concept was used by Eudemus, who regarded an angle as a deviation from a straight line; the second by Carpus of Antioch, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third concept, although his definitions of right, acute, and obtuse angles are certainly quantitative.[3]

Contents 1	Identifying angles 2	Types of angles 2.1	Individual angles 2.2	Equivalence angle pairs 2.3	Vertical and adjacent angle pairs 2.4	Combining angle pairs 2.5	Polygon-related angles 2.6	Plane-related angles 3	Measuring angles 3.1	Angle addition postulate 3.2	Units 3.3	Positive and negative angles 3.4	Alternative ways of measuring the size of an angle 3.5	Astronomical approximations 4	Angles between curves 5	Bisecting and trisecting angles 6	Dot product and generalisations 6.1	Inner product 6.2	Angles between subspaces 6.3	Angles in Riemannian geometry 6.4	Hyperbolic angle 7	Angles in geography and astronomy 8	See also 9	Notes 10	References 11	Bibliography 12	External links Identifying angles In mathematical expressions, it is common to use Greek letters (α, β, γ, θ, φ, . . . ) as variables denoting the size of some angle[4] (to avoid confusion with its other meaning, the symbol π is typically not used for this purpose). Lower case Roman letters (a, b, c, . . . ) are also used, as are upper case Roman letters in the context of polygons. See the figures in this article for examples.

In geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB and AC (i.e. the lines from point A to point B and point A to point C) is denoted ∠BAC (in Unicode U+2220 ∠ ANGLE) or {\displaystyle {\widehat {\rm {BAC}}}}{\displaystyle {\widehat {\rm {BAC}}}}. Where there is no risk of confusion, the angle may sometimes be referred to simply by its vertex (in this case "angle A").

Potentially, an angle denoted as, say, ∠BAC, might refer to any of four angles: the clockwise angle from B to C, the anticlockwise angle from B to C, the clockwise angle from C to B, or the anticlockwise angle from C to B, where the direction in which the angle is measured determines its sign (see Positive and negative angles). However, in many geometrical situations, it is obvious from context that the positive angle less than or equal to 180 degrees is meant, in which case no ambiguity arises. Otherwise, a convention may be adopted so that ∠BAC always refers to the anticlockwise (positive) angle from B to C, and ∠CAB the anticlockwise (positive) angle from C to B.

Types of angles "Oblique angle" redirects here. For the cinematographic technique, see Dutch angle. Individual angles There is some common terminology for angles, whose measure is always non-negative (see #Positive and negative angles):[5][6]

An angle equal to 0° or not turned is called a zero angle. Angles smaller than a right angle (less than 90°) are called acute angles ("acute" meaning "sharp"). An angle equal to 1 / 4 turn (90° or π / 2 radians) is called a right angle. Two lines that form a right angle are said to be normal, orthogonal, or perpendicular. Angles larger than a right angle and smaller than a straight angle (between 90° and 180°) are called obtuse angles ("obtuse" meaning "blunt"). An angle equal to 1 / 2 turn (180° or π radians) is called a straight angle. Angles larger than a straight angle but less than 1 turn (between 180° and 360°) are called reflex angles. An angle equal to 1 turn (360° or 2π radians) is called a full angle, complete angle, round angle or a perigon. Angles that are not right angles or a multiple of a right angle are called oblique angles. The names, intervals, and measured units are shown in the table below:

Right angle

Acute (a), obtuse (b), and straight (c) angles. The acute and obtuse angles are also known as oblique angles.

Reflex angle Name 	zero	acute	right angle	obtuse	straight	reflex	perigon Units	Interval Turns 	0	(0,  1 / 4 ) 1 / 4 ( 1 / 4, 1 / 2 )	1 / 2 ( 1 / 2 ,  1)	1 Radians	0	(0, 1 / 2 π) 1 / 2 π	( 1 / 2 π, π)	π	(π, 2π)	2π Degrees 	0°	(0, 90)°	90°	(90, 180)°	180°	(180, 360)°	360° Gons 	0g	(0, 100)g	100g	(100, 200)g	200g	(200, 400)g	400g Equivalence angle pairs Angles that have the same measure (i.e. the same magnitude) are said to be equal or congruent. An angle is defined by its measure and is not dependent upon the lengths of the sides of the angle (e.g. all right angles are equal in measure). Two angles which share terminal sides, but differ in size by an integer multiple of a turn, are called coterminal angles. A reference angle is the acute version of any angle determined by repeatedly subtracting or adding straight angle ( 1 / 2 turn, 180°, or π radians), to the results as necessary, until the magnitude of result is an acute angle, a value between 0 and 1 / 4 turn, 90°, or π / 2 radians. For example, an angle of 30 degrees has a reference angle of 30 degrees, and an angle of 150 degrees also has a reference angle of 30 degrees (180–150). An angle of 750 degrees has a reference angle of 30 degrees (750–720).[7] Vertical and adjacent angle pairs

Angles A and B are a pair of vertical angles; angles C and D are a pair of vertical angles. When two straight lines intersect at a point, four angles are formed. Pairwise these angles are named according to their location relative to each other.

A pair of angles opposite each other, formed by two intersecting straight lines that form an "X"-like shape, are called vertical angles or opposite angles or vertically opposite angles. They are abbreviated as vert. opp. ∠s.[8] The equality of vertically opposite angles is called the vertical angle theorem. Eudemus of Rhodes attributed the proof to Thales of Miletus.[9][10] The proposition showed that since both of a pair of vertical angles are supplementary to both of the adjacent angles, the vertical angles are equal in measure. According to a historical note,[10] when Thales visited Egypt, he observed that whenever the Egyptians drew two intersecting lines, they would measure the vertical angles to make sure that they were equal. Thales concluded that one could prove that all vertical angles are equal, if one accepted some general notions such as: All straight angles are equal. Equals added to equals are equal. Equals subtracted from equals are equal. When two adjacent angles form a straight line, they are supplementary. Therefore, if we assume that the measure of angle A equals x, then the measure of angle C would be 180 − x. Similarly, the measure of angle D would be 180 − x. Both angle C and angle D have measures equal to 180 − x and are congruent. Since angle B is supplementary to both angles C and D, either of these angle measures may be used to determine the measure of Angle B. Using the measure of either angle C or angle D, we find the measure of angle B to be 180 − (180 − x) = 180 − 180 + x = x. Therefore, both angle A and angle B have measures equal to x and are equal in measure.

Angles A and B are adjacent. Adjacent angles, often abbreviated as adj. ∠s, are angles that share a common vertex and edge but do not share any interior points. In other words, they are angles that are side by side, or adjacent, sharing an "arm". Adjacent angles which sum to a right angle, straight angle or full angle are special, and are respectively called complementary, supplementary and explementary angles (see "Combine angle pairs" below). A transversal is a line that intersects a pair of (often parallel) lines, and is associated with alternate interior angles, corresponding angles, interior angles, and exterior angles.[11]

Combining angle pairs There are three special angle pairs which involve the summation of angles:

The complementary angles a and b (b is the complement of a, and a is the complement of b). Complementary angles are angle pairs whose measures sum to one right angle ( 1 / 4 turn, 90°, or π / 2 radians).[12] If the two complementary angles are adjacent, their non-shared sides form a right angle. In Euclidean geometry, the two acute angles in a right triangle are complementary, because the sum of internal angles of a triangle is 180 degrees, and the right angle itself accounts for 90 degrees. The adjective complementary is from Latin complementum, associated with the verb complere, "to fill up". An acute angle is "filled up" by its complement to form a right angle. The difference between an angle and a right angle is termed the complement of the angle.[13] If angles A and B are complementary, the following relationships hold: {\displaystyle {\begin{aligned}&\sin ^{2}A+\sin ^{2}B=1&&\cos ^{2}A+\cos ^{2}B=1\\[3pt]&\tan A=\cot B&&\sec A=\csc B\end{aligned}}}{\displaystyle {\begin{aligned}&\sin ^{2}A+\sin ^{2}B=1&&\cos ^{2}A+\cos ^{2}B=1\\[3pt]&\tan A=\cot B&&\sec A=\csc B\end{aligned}}} (The tangent of an angle equals the cotangent of its complement and its secant equals the cosecant of its complement.) The prefix "co-" in the names of some trigonometric ratios refers to the word "complementary".

The angles a and b are supplementary angles. Two angles that sum to a straight angle ( 1 / 2 turn, 180°, or π radians) are called supplementary angles.[14] If the two supplementary angles are adjacent (i.e. have a common vertex and share just one side), their non-shared sides form a straight line. Such angles are called a linear pair of angles.[15] However, supplementary angles do not have to be on the same line, and can be separated in space. For example, adjacent angles of a parallelogram are supplementary, and opposite angles of a cyclic quadrilateral (one whose vertices all fall on a single circle) are supplementary. If a point P is exterior to a circle with center O, and if the tangent lines from P touch the circle at points T and Q, then ∠TPQ and ∠TOQ are supplementary. The sines of supplementary angles are equal. Their cosines and tangents (unless undefined) are equal in magnitude but have opposite signs. In Euclidean geometry, any sum of two angles in a triangle is supplementary to the third, because the sum of internal angles of a triangle is a straight angle.

Sum of two explementary angles is a complete angle. Two angles that sum to a complete angle (1 turn, 360°, or 2π radians) are called explementary angles or conjugate angles. The difference between an angle and a complete angle is termed the explement of the angle or conjugate of an angle. Polygon-related angles

Internal and external angles. An angle that is part of a simple polygon is called an interior angle if it lies on the inside of that simple polygon. A simple concave polygon has at least one interior angle that is a reflex angle. In Euclidean geometry, the measures of the interior angles of a triangle add up to π radians, 180°, or 1 / 2 turn; the measures of the interior angles of a simple convex quadrilateral add up to 2π radians, 360°, or 1 turn. In general, the measures of the interior angles of a simple convex polygon with n sides add up to (n − 2)π radians, or 180(n − 2) degrees, (2n − 4) right angles, or ( n / 2 − 1) turn. The supplement of an interior angle is called an exterior angle, that is, an interior angle and an exterior angle form a linear pair of angles. There are two exterior angles at each vertex of the polygon, each determined by extending one of the two sides of the polygon that meet at the vertex; these two angles are vertical angles and hence are equal. An exterior angle measures the amount of rotation one has to make at a vertex to trace out the polygon.[16] If the corresponding interior angle is a reflex angle, the exterior angle should be considered negative. Even in a non-simple polygon it may be possible to define the exterior angle, but one will have to pick an orientation of the plane (or surface) to decide the sign of the exterior angle measure. In Euclidean geometry, the sum of the exterior angles of a simple convex polygon will be one full turn (360°). The exterior angle here could be called a supplementary exterior angle. Exterior angles are commonly used in Logo Turtle programs when drawing regular polygons. In a triangle, the bisectors of two exterior angles and the bisector of the other interior angle are concurrent (meet at a single point).[17]:p. 149 In a triangle, three intersection points, each of an external angle bisector with the opposite extended side, are collinear.[17]:p. 149 In a triangle, three intersection points, two of them between an interior angle bisector and the opposite side, and the third between the other exterior angle bisector and the opposite side extended, are collinear.[17]:p. 149 Some authors use the name exterior angle of a simple polygon to simply mean the explement exterior angle (not supplement!) of the interior angle.[18] This conflicts with the above usage. Plane-related angles The angle between two planes (such as two adjacent faces of a polyhedron) is called a dihedral angle.[13] It may be defined as the acute angle between two lines normal to the planes. The angle between a plane and an intersecting straight line is equal to ninety degrees minus the angle between the intersecting line and the line that goes through the point of intersection and is normal to the plane. Measuring angles

It has been suggested that Angular unit be merged into this article. (Discuss) Proposed since May 2020. The size of a geometric angle is usually characterized by the magnitude of the smallest rotation that maps one of the rays into the other. Angles that have the same size are said to be equal or congruent or equal in measure.

In some contexts, such as identifying a point on a circle or describing the orientation of an object in two dimensions relative to a reference orientation, angles that differ by an exact multiple of a full turn are effectively equivalent. In other contexts, such as identifying a point on a spiral curve or describing the cumulative rotation of an object in two dimensions relative to a reference orientation, angles that differ by a non-zero multiple of a full turn are not equivalent.

The measure of angle θ (in radians) is the quotient of s and r. In order to measure an angle θ, a circular arc centered at the vertex of the angle is drawn, e.g. with a pair of compasses. The ratio of the length s of the arc by the radius r of the circle is the measure of the angle in radians.

The measure of the angle in another angular unit is then obtained by multiplying its measure in radians by the scaling factor k / 2π , where k is the measure of a complete turn in the chosen unit (for example 360 for degrees or 400 for gradians):

{\displaystyle \theta =k{\frac {s}{2\pi r}}.}\theta =k{\frac {s}{2\pi r}}. The value of θ thus defined is independent of the size of the circle: if the length of the radius is changed then the arc length changes in the same proportion, so the ratio s/r is unaltered. (Proof. The formula above can be rewritten as k = θr / s . One turn, for which θ = n units, corresponds to an arc equal in length to the circle's circumference, which is 2πr, so s = 2πr. Substituting n for θ and 2πr for s in the formula, results in k = nr / 2πr = n / 2π .) [nb 1]

Angle addition postulate The angle addition postulate states that if B is in the interior of angle AOC, then

{\displaystyle m\angle AOC=m\angle AOB+m\angle BOC}m\angle AOC=m\angle AOB+m\angle BOC The measure of the angle AOC is the sum of the measure of angle AOB and the measure of angle BOC. In this postulate it does not matter in which unit the angle is measured as long as each angle is measured in the same unit.

Units See also: Angular unit Units used to represent angles are listed below in descending magnitude order. Of these units, the degree and the radian are by far the most commonly used. Angles expressed in radians are dimensionless for the purposes of dimensional analysis.

Most units of angular measurement are defined such that one turn (i.e. one full circle) is equal to n units, for some whole number n. The two exceptions are the radian and the diameter part.

Turn (n = 1) The turn, also cycle, full circle, revolution, and rotation, is complete circular movement or measure (as to return to the same point) with circle or ellipse. A turn is abbreviated τ, cyc, rev, or rot depending on the application, but in the acronym rpm (revolutions per minute), just r is used. A turn of n units is obtained by setting k = 1 / 2π in the formula above. The equivalence of 1 turn is 360°, 2π rad, 400 grad, and 4 right angles. The symbol τ can also be used as a mathematical constant to represent 2π radians. Used in this way (k = τ / 2π ) allows for radians to be expressed as a fraction of a turn. For example, half a turn is τ / 2 = π. Quadrant (n = 4) The quadrant is 1 / 4 of a turn, i.e. a right angle. It is the unit used in Euclid's Elements. 1 quad. = 90° = π / 2 rad = 1 / 4 turn = 100 grad. In German the symbol ∟ has been used to denote a quadrant. Sextant (n = 6) The sextant (angle of the equilateral triangle) is 1 / 6 of a turn. It was the unit used by the Babylonians,[20][21] and is especially easy to construct with ruler and compasses. The degree, minute of arc and second of arc are sexagesimal subunits of the Babylonian unit. 1 Babylonian unit = 60° = π/3 rad ≈ 1.047197551 rad.

θ = s/r rad = 1 rad. Radian (n = 2π =  6.283 . . . ) The radian is the angle subtended by an arc of a circle that has the same length as the circle's radius. The case of radian for the formula given earlier, a radian of n = 2π units is obtained by setting k = 2π / 2π = 1. One turn is 2π radians, and one radian is 180 / π degrees, or about 57.2958 degrees. The radian is abbreviated rad, though this symbol is often omitted in mathematical texts, where radians are assumed unless specified otherwise. When radians are used angles are considered as dimensionless. The radian is used in virtually all mathematical work beyond simple practical geometry, due, for example, to the pleasing and "natural" properties that the trigonometric functions display when their arguments are in radians. The radian is the (derived) unit of angular measurement in the SI system. Clock position (n = 12) A clock position is the relative direction of an object described using the analogy of a 12-hour clock. One imagines a clock face lying either upright or flat in front of oneself, and identifies the twelve hour markings with the directions in which they point. Hour angle (n = 24) The astronomical hour angle is 1 / 24 of a turn. As this system is amenable to measuring objects that cycle once per day (such as the relative position of stars), the sexagesimal subunits are called minute of time and second of time. These are distinct from, and 15 times larger than, minutes and seconds of arc. 1 hour = 15° = π / 12 rad = 1 / 6 quad. = 1 / 24 turn = 16+ 2 / 3 grad. (Compass) point or wind (n = 32) The point, used in navigation, is 1 / 32 of a turn. 1 point = 1 / 8 of a right angle = 11.25° = 12.5 grad. Each point is subdivided in four quarter-points so that 1 turn equals 128 quarter-points. Hexacontade (n = 60) The hexacontade is a unit of 6° that Eratosthenes used, so that a whole turn was divided into 60 units. Pechus (n = 144–180) The pechus was a Babylonian unit equal to about 2° or 2+ 1 / 2 °. Binary degree (n = 256) The binary degree, also known as the binary radian (or brad), is 1 / 256 of a turn.[22] The binary degree is used in computing so that an angle can be efficiently represented in a single byte (albeit to limited precision). Other measures of angle used in computing may be based on dividing one whole turn into 2n equal parts for other values of n.[23] Degree (n = 360) The degree, denoted by a small superscript circle (°), is 1/360 of a turn, so one turn is 360°. The case of degrees for the formula given earlier, a degree of n = 360° units is obtained by setting k = 360° / 2π . One advantage of this old sexagesimal subunit is that many angles common in simple geometry are measured as a whole number of degrees. Fractions of a degree may be written in normal decimal notation (e.g. 3.5° for three and a half degrees), but the "minute" and "second" sexagesimal subunits of the "degree-minute-second" system are also in use, especially for geographical coordinates and in astronomy and ballistics. Diameter part (n = 376.99 . . . ) The diameter part (occasionally used in Islamic mathematics) is 1 / 60 radian. One "diameter part" is approximately 0.95493°. There are about 376.991 diameter parts per turn. Grad (n = 400) The grad, also called grade, gradian, or gon, is 1 / 400 of a turn, so a right angle is 100 grads.[4] It is a decimal subunit of the quadrant. A kilometre was historically defined as a centi-grad of arc along a great circle of the Earth, so the kilometer is the decimal analog to the sexagesimal nautical mile. The grad is used mostly in triangulation. Milliradian The milliradian (mil or mrad) is defined as a thousandth of a radian, which means that a rotation of one turn consists of 2000π mil (or approximately 6283.185... mil), and almost all scope sights for firearms are calibrated to this definition. In addition there are three other derived definitions used for artillery and navigation which are approximately equal to a milliradian. Under these three other definitions one turn makes up for exactly 6000, 6300 or 6400 mils, which equals spanning the range from 0.05625 to 0.06 degrees (3.375 to 3.6 minutes). In comparison, the true milliradian is approximately 0.05729578... degrees (3.43775... minutes). One "NATO mil" is defined as 1 / 6400 of a circle. Just like with the true milliradian, each of the other definitions exploits the mil's handby property of subtensions, i.e. that the value of one milliradian approximately equals the angle subtended by a width of 1 meter as seen from 1 km away ( 2π / 6400 = 0.0009817... ≈ 1 / 1000 ). Minute of arc (n = 21,600) The minute of arc (or MOA, arcminute, or just minute) is 1 / 60 of a degree = 1 / 21,600 turn. It is denoted by a single prime ( ′ ). For example, 3° 30′ is equal to 3 × 60 + 30 = 210 minutes or 3 + 30 / 60 = 3.5 degrees. A mixed format with decimal fractions is also sometimes used, e.g. 3° 5.72′ = 3 + 5.72 / 60 degrees. A nautical mile was historically defined as a minute of arc along a great circle of the Earth. Second of arc (n = 1,296,000) The second of arc (or arcsecond, or just second) is 1 / 60 of a minute of arc and 1 / 3600 of a degree. It is denoted by a double prime ( ″ ). For example, 3° 7′ 30″ is equal to 3 + 7 / 60 + 30 / 3600 degrees, or 3.125 degrees. Milliarcsecond (n = 1,296,000,000) mas Microarcsecond (n = 1,296,000,000,000) µas Positive and negative angles Although the definition of the measurement of an angle does not support the concept of a negative angle, it is frequently useful to impose a convention that allows positive and negative angular values to represent orientations and/or rotations in opposite directions relative to some reference.

In a two-dimensional Cartesian coordinate system, an angle is typically defined by its two sides, with its vertex at the origin. The initial side is on the positive x-axis, while the other side or terminal side is defined by the measure from the initial side in radians, degrees, or turns. With positive angles representing rotations toward the positive y-axis and negative angles representing rotations toward the negative y-axis. When Cartesian coordinates are represented by standard position, defined by the x-axis rightward and the y-axis upward, positive rotations are anticlockwise and negative rotations are clockwise.

In many contexts, an angle of −θ is effectively equivalent to an angle of "one full turn minus θ". For example, an orientation represented as −45° is effectively equivalent to an orientation represented as 360° − 45° or 315°. Although the final position is the same, a physical rotation (movement) of −45° is not the same as a rotation of 315° (for example, the rotation of a person holding a broom resting on a dusty floor would leave visually different traces of swept regions on the floor).

In three-dimensional geometry, "clockwise" and "anticlockwise" have no absolute meaning, so the direction of positive and negative angles must be defined relative to some reference, which is typically a vector passing through the angle's vertex and perpendicular to the plane in which the rays of the angle lie.

In navigation, bearings or azimuth are measured relative to north. By convention, viewed from above, bearing angles are positive clockwise, so a bearing of 45° corresponds to a north-east orientation. Negative bearings are not used in navigation, so a north-west orientation corresponds to a bearing of 315°.

Alternative ways of measuring the size of an angle There are several alternatives to measuring the size of an angle by the angle of rotation. The grade of a slope, or gradient is equal to the tangent of the angle, or sometimes (rarely) the sine. A gradient is often expressed as a percentage. For very small values (less than 5%), the grade of a slope is approximately the measure of the angle in radians.

In rational geometry the spread between two lines is defined as the square of the sine of the angle between the lines. As the sine of an angle and the sine of its supplementary angle are the same, any angle of rotation that maps one of the lines into the other leads to the same value for the spread between the lines.

Astronomical approximations Main article: Angular diameter Astronomers measure angular separation of objects in degrees from their point of observation.

0.5° is approximately the width of the sun or moon. 1° is approximately the width of a little finger at arm's length. 10° is approximately the width of a closed fist at arm's length. 20° is approximately the width of a handspan at arm's length. These measurements clearly depend on the individual subject, and the above should be treated as rough rule of thumb approximations only.

Angles between curves

The angle between the two curves at P is defined as the angle between the tangents A and B at P. The angle between a line and a curve (mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the tangents at the point of intersection. Various names (now rarely, if ever, used) have been given to particular cases:—amphicyrtic (Gr. ἀμφί, on both sides, κυρτός, convex) or cissoidal (Gr. κισσός, ivy), biconvex; xystroidal or sistroidal (Gr. ξυστρίς, a tool for scraping), concavo-convex; amphicoelic (Gr. κοίλη, a hollow) or angulus lunularis, biconcave.[24]

Bisecting and trisecting angles Main articles: Bisection § Angle bisector, and Angle trisection The ancient Greek mathematicians knew how to bisect an angle (divide it into two angles of equal measure) using only a compass and straightedge, but could only trisect certain angles. In 1837 Pierre Wantzel showed that for most angles this construction cannot be performed.

Dot product and generalisations In the Euclidean space, the angle θ between two Euclidean vectors u and v is related to their dot product and their lengths by the formula

{\displaystyle \mathbf {u} \cdot \mathbf {v} =\cos(\theta )\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}{\displaystyle \mathbf {u} \cdot \mathbf {v} =\cos(\theta )\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.} This formula supplies an easy method to find the angle between two planes (or curved surfaces) from their normal vectors and between skew lines from their vector equations.

Inner product To define angles in an abstract real inner product space, we replace the Euclidean dot product ( · ) by the inner product {\displaystyle \langle \cdot ,\cdot \rangle }\langle \cdot ,\cdot \rangle, i.e.

{\displaystyle \langle \mathbf {u} ,\mathbf {v} \rangle =\cos(\theta )\ \left\|\mathbf {u} \right\|\ \left\|\mathbf {v} \right\|.}\langle \mathbf {u} ,\mathbf {v} \rangle =\cos(\theta )\ \left\|\mathbf {u} \right\|\ \left\|\mathbf {v} \right\|. In a complex inner product space, the expression for the cosine above may give non-real values, so it is replaced with

{\displaystyle \operatorname {Re} \left(\langle \mathbf {u} ,\mathbf {v} \rangle \right)=\cos(\theta )\ \left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}{\displaystyle \operatorname {Re} \left(\langle \mathbf {u} ,\mathbf {v} \rangle \right)=\cos(\theta )\ \left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.} or, more commonly, using the absolute value, with

{\displaystyle \left|\langle \mathbf {u} ,\mathbf {v} \rangle \right|=|\cos(\theta )|\ \left\|\mathbf {u} \right\|\ \left\|\mathbf {v} \right\|.}\left|\langle \mathbf {u} ,\mathbf {v} \rangle \right|=|\cos(\theta )|\ \left\|\mathbf {u} \right\|\ \left\|\mathbf {v} \right\|. The latter definition ignores the direction of the vectors and thus describes the angle between one-dimensional subspaces {\displaystyle \operatorname {span} (\mathbf {u} )}\operatorname {span} (\mathbf {u} ) and {\displaystyle \operatorname {span} (\mathbf {v} )}\operatorname {span} (\mathbf {v} ) spanned by the vectors {\displaystyle \mathbf {u} }\mathbf {u} and {\displaystyle \mathbf {v} }\mathbf {v}  correspondingly.

Angles between subspaces The definition of the angle between one-dimensional subspaces {\displaystyle \operatorname {span} (\mathbf {u} )}\operatorname {span} (\mathbf {u} ) and {\displaystyle \operatorname {span} (\mathbf {v} )}\operatorname {span} (\mathbf {v} ) given by

{\displaystyle \left|\langle \mathbf {u} ,\mathbf {v} \rangle \right|=|\cos(\theta )|\left\|\mathbf {u} \right\|\ \left\|\mathbf {v} \right\|}{\displaystyle \left|\langle \mathbf {u} ,\mathbf {v} \rangle \right|=|\cos(\theta )|\left\|\mathbf {u} \right\|\ \left\|\mathbf {v} \right\|} in a Hilbert space can be extended to subspaces of any finite dimensions. Given two subspaces {\displaystyle {\mathcal {U}}}{\mathcal {U}}, {\displaystyle {\mathcal {W}}}{\mathcal {W}} with {\displaystyle \dim({\mathcal {U}}):=k\leq \dim({\mathcal {W}}):=l}{\displaystyle \dim({\mathcal {U}}):=k\leq \dim({\mathcal {W}}):=l}, this leads to a definition of {\displaystyle k}k angles called canonical or principal angles between subspaces.

Angles in Riemannian geometry In Riemannian geometry, the metric tensor is used to define the angle between two tangents. Where U and V are tangent vectors and gij are the components of the metric tensor G,

{\displaystyle \cos \theta ={\frac {g_{ij}U^{i}V^{j}}{\sqrt {\left|g_{ij}U^{i}U^{j}\right|\left|g_{ij}V^{i}V^{j}\right|}}}.}{\displaystyle \cos \theta ={\frac {g_{ij}U^{i}V^{j}}{\sqrt {\left|g_{ij}U^{i}U^{j}\right|\left|g_{ij}V^{i}V^{j}\right|}}}.} Hyperbolic angle A hyperbolic angle is an argument of a hyperbolic function just as the circular angle is the argument of a circular function. The comparison can be visualized as the size of the openings of a hyperbolic sector and a circular sector since the areas of these sectors correspond to the angle magnitudes in each case. Unlike the circular angle, the hyperbolic angle is unbounded. When the circular and hyperbolic functions are viewed as infinite series in their angle argument, the circular ones are just alternating series forms of the hyperbolic functions. This weaving of the two types of angle and function was explained by Leonhard Euler in Introduction to the Analysis of the Infinite.

Angles in geography and astronomy In geography, the location of any point on the Earth can be identified using a geographic coordinate system. This system specifies the latitude and longitude of any location in terms of angles subtended at the centre of the Earth, using the equator and (usually) the Greenwich meridian as references.

In astronomy, a given point on the celestial sphere (that is, the apparent position of an astronomical object) can be identified using any of several astronomical coordinate systems, where the references vary according to the particular system. Astronomers measure the angular separation of two stars by imagining two lines through the centre of the Earth, each intersecting one of the stars. The angle between those lines can be measured, and is the angular separation between the two stars.

In both geography and astronomy, a sighting direction can be specified in terms of a vertical angle such as altitude /elevation with respect to the horizon as well as the azimuth with respect to north.

Astronomers also measure the apparent size of objects as an angular diameter. For example, the full moon has an angular diameter of approximately 0.5°, when viewed from Earth. One could say, "The Moon's diameter subtends an angle of half a degree." The small-angle formula can be used to convert such an angular measurement into a distance/size ratio.

See also Angle bisector Angular velocity Argument (complex analysis) Astrological aspect Central angle Clock angle problem Dihedral angle Exterior angle theorem Golden angle Great circle distance Inscribed angle Irrational angle Phase (waves) Protractor Solid angle for a concept of angle in three dimensions. Spherical angle Transcendent angle Trisection Zenith angle Notes This approach requires however an additional proof that the measure of the angle does not change with changing radius r, in addition to the issue of "measurement units chosen." A smoother approach is to measure the angle by the length of the corresponding unit circle arc. Here "unit" can be chosen to be dimensionless in the sense that it is the real number 1 associated with the unit segment on the real line. See Radoslav M. Dimitrić for instance.[19] References Sidorov 2001 Slocum 2007 Chisholm 1911; Heiberg 1908, pp. 177–178 "Compendium of Mathematical Symbols". Math Vault. 2020-03-01. Retrieved 2020-08-17. "Angles - Acute, Obtuse, Straight and Right". www.mathsisfun.com. Retrieved 2020-08-17. Weisstein, Eric W. "Angle". mathworld.wolfram.com. Retrieved 2020-08-17. "Mathwords: Reference Angle". www.mathwords.com. Archived from the original on 23 October 2017. Retrieved 26 April 2018. Wong & Wong 2009, pp. 161–163 Euclid. The Elements. Proposition I:13. Shute, Shirk & Porter 1960, pp. 25–27. Jacobs 1974, p. 255. "Complementary Angles". www.mathsisfun.com. Retrieved 2020-08-17. Chisholm 1911 "Supplementary Angles". www.mathsisfun.com. Retrieved 2020-08-17. Jacobs 1974, p. 97. Henderson & Taimina 2005, p. 104. Johnson, Roger A. Advanced Euclidean Geometry, Dover Publications, 2007. D. Zwillinger, ed. (1995), CRC Standard Mathematical Tables and Formulae, Boca Raton, FL: CRC Press, p. 270 as cited in Weisstein, Eric W. "Exterior Angle". MathWorld. Dimitrić, Radoslav M. (2012). "On Angles and Angle Measurements" (PDF). The Teaching of Mathematics. XV (2): 133–140. Archived (PDF) from the original on 2019-01-17. Retrieved 2019-08-06. Jeans, James Hopwood (1947). The Growth of Physical Science. CUP Archive. p. 7. Murnaghan, Francis Dominic (1946). Analytic Geometry. p. 2. "ooPIC Programmer's Guide - Chapter 15: URCP". ooPIC Manual & Technical Specifications - ooPIC Compiler Ver 6.0. Savage Innovations, LLC. 2007 [1997]. Archived from the original on 2008-06-28. Retrieved 2019-08-05. Hargreaves, Shawn. "Angles, integers, and modulo arithmetic". blogs.msdn.com. Archived from the original on 2019-06-30. Retrieved 2019-08-05. Chisholm 1911; Heiberg 1908, p. 178 Bibliography Henderson, David W.; Taimina, Daina (2005), Experiencing Geometry / Euclidean and Non-Euclidean with History (3rd ed.), Pearson Prentice Hall, p. 104, ISBN 978-0-13-143748-7 Heiberg, Johan Ludvig (1908), Heath, T. L. (ed.), Euclid, The Thirteen Books of Euclid's Elements, 1, Cambridge: Cambridge University Press. Sidorov, L. A. (2001) [1994], "Angle", Encyclopedia of Mathematics, EMS Press Jacobs, Harold R. (1974), Geometry, W. H. Freeman, pp. 97, 255, ISBN 978-0-7167-0456-0 Slocum, Jonathan (2007), Preliminary Indo-European lexicon — Pokorny PIE data, University of Texas research department: linguistics research center, retrieved 2 Feb 2010 Shute, William G.; Shirk, William W.; Porter, George F. (1960), Plane and Solid Geometry, American Book Company, pp. 25–27 Wong, Tak-wah; Wong, Ming-sim (2009), "Angles in Intersecting and Parallel Lines", New Century Mathematics, 1B (1 ed.), Hong Kong: Oxford University Press, pp. 161–163, ISBN 978-0-19-800177-5 This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911), "Angle", Encyclopædia Britannica, 2 (11th ed.), Cambridge University Press, p. 14

ENGLISH LANGUAGE

Verbs

Transitive and intransitive Verbs Verbs can be classified into transitive and intransitive. A transitive verb needs an object while the intransitive does not. Many verbs can be used as both transitive and intransitive depending on how they are used in a sentence. Transitive verb A transitive verb has to be an action verb, and it must have an object. Without an object, it does not convey a complete meaning. Example: He bought. The question inevitably arises: What did he buy? No one in the world knows the answer to this question as there is no direct object to tell us what he bought. The meaning becomes clear when an object is added: He bought a cake. Now every one of us knows what he bought. The subject he performs the action bought, which is the transitive verb acting upon the object of the sentence cake This following example shows each transitive verb (in capital letters) having a direct object (brackets) to complete the sentence. If it doesn’t have a direct object, it makes the sentence meaningless. Examples: I have to CATCH the earliest (train )tomorrow. We agreed to SETTLE the (lawsuit) out of court. I PUSHED the (button) and nothing happened. They PICKED him as the (captain). I WROTE the (number) somewhere. A transitive verb may take an indirect object. An indirect object is someone or something to whom/which or for whom/which the action is carried out. (a) Example: He BOUGHT (her)a cake. / He bought a cake for her. (b) Example: She IS READING (grandma) a fairy tale. / She is reading a fairy tale to grandma. In the Example (a), the indirect object is her as it is for her that the cake was bought. In the Example (b), the indirect object is grandma as it is to her that the fairy tale is read. The indirect object usually comes before the direct object as happened in the first sentences of the two examples. Intransitive verb An intransitive verb does not need an object to make the sentence's meaning clear. It is enough to complete a sentence without an object as the meaning of the sentence is not affected. The following examples show the intransitive verbs in Capital letters. Examples: She SMILES The dog IS BARKING. Their plane has already TAKEN OFF. The people next door are ARGUING loudly. She HAS BEEN SNEEZING since this morning. Transitive and Intransitive verbs Many verbs can be both transitive and intransitive. When a transitive verb is used intransitively, the meaning changes. In the following examples, the verbs are in Capital letters and the direct objects are in brackets. Examples: Transitive verb: It is better we EAT (something) before we go. Intransitive verb: Our parents like to EAT OUT on Sundays. Transitive verb: They PLAYED (hide-and-seek) yesterday. Intransitive verb: The children PLAYED in the park. Transitive verb: The (fat boy) cannot TOUCH his toes. Intransitive verb: The sign says, “Please don't TOUCH". Transitive verb: When she heard what happened, she CRIED tears of joy. Intransitive verb: Someone is crying loudly. Intransitive verb followed by adverb or prepositional phrase Since an intransitive verb cannot take an object, it can never be followed by a noun. But it can be followed by an adverb or prepositional phrase, or both. Examples: The family lives upstairs. (Intransitive verb lives followed by adverb upstairs.)

Note the followings before proceeding to identify the sentences that have a transitive or intransitive verb: The verb that needs an object to make its meaning clear or complete is called a Transitive Verb. The object can be a noun or a pronoun. The Intransitive Verb does not need an object to complete its meaning. Each of the following sentences has a verb that is either transitive or intransitive. Choose the correct answer – transitive or intransitive – as given at the end of the sentence. 1. She saw a ghost. (Transitive/Intransitive) 1. She saw a ghost. (Transitive verb – object: ghost) 2. My vase dropped and broke into pieces. (Transitive/Intransitive) 2. My vase dropped and broke into pieces. (Intransitive verb – no object) 3. Finally, one of them found the golf ball. (Transitive/Intransitive) 3. Finally, one of them found the golf ball. (Transitive verb – object: golf ball) 4. A big cat is chasing a small dog. (Transitive/Intransitive) 4. A big cat is chasing a small dog. (Transitive verb – object: dog) 5. He used to come here every day. (Transitive/Intransitive) 5. He used to come here every day. (Intransitive verb – no object) 6. He landed one very big fish. (Transitive/Intransitive) 6. He landed one very big fish. (Transitive verb – object: fish) 7. You always drink and talk endlessly. (Transitive/Intransitive) 7. You always drink and talk endlessly. (Intransitive verb – no object) 8. Tom likes to read the latest books. (Transitive/Intransitive) 8. Tom likes to read the latest books. (Transitive verb – object: books) 9. The tourists visited the war museum. (Transitive/Intransitive) 9. The tourists visited the war museum. (Transitive verb – object: museum) 10. She can cry very loudly. (Transitive/Intransitive) 10. She can cry very loudly. (Intransitive verb – no object) 11. All her valuables were safely locked away. (Transitive/Intransitive) whom you can be as easy as He’s 11. All her valuables were safely locked away. (Intransitive verb – no object) 12. Every evening Tommy plays football. (Transitive/Intransitive) 12. Every evening Tommy plays football. (Transitive verb – object: football) years 1989 a sort of my business man who is alive and 13. Some grandmothers can still knit very well. (Transitive/Intransitive) 13. Some grandmother can still knit very well. (Intransitive verb – no object) ruling are looking more than two dozen name will be an easy for all his 14.the as the only thing once you know if my for the whole thing was that he uses various units She sang a tribal folk song. (Transitive/Intransitive) 14. She sang a tribal folk song. (Transitive verb – object: tribal folk song) 15. A species of beetle could soon disappear forever. (Transitive/Intransitive) 15. A species of beetle could soon disappear forever. (Intransitive verb – no object) 2.2.1 Linking Verbs

ICT

HUMANWARE/PEOPLEWARE Computer people/Humanware refers to as the end users of the computer system. They are the people working with computer system. Computer humanware is divided by into two namely: 1.	Computer professionals 2.	Computer users

COMPUTER PROFESSIONALS Computer professional are the individuals that have obtained sufficient education and training in the field of computer Science. Examples of computer professionals include i.	Computer Engineers ii. Computer Analyst iii. Computer programmer iv. Computer system administrator v.	Computer scientist vi. Data processing managers vii. Computer Educators viii. Computer operator

COMPUTER USERS Computer users are the people who are not computer professionals but which make use of computers in doing their jobs. They include bankers, doctors, architect, accountants, teacher etc. SOME FUNCTIONS OF COMPUTER PROFESSIONALS 1.	Produces flowcharts for the programmer 2.	Responsible for writing programs by using the flowcharts produced by the system analyst. 3.	Responsible for data preparation. 4.	He inputs data into the computer 5.	Responsible for keeping books, manuals and other documentations. 6.	Maintenance of the computer and peripherals so that they can be in good working conditions. 7.	To repair or replace the defective components in the computer. 8.	To periodically cleans the computer and the sensitive inner parts. 9.	To monitor the computer environment and specifies he ideal conditions. 10.	To input data into the computer etc.

QUALITIES OF GOOD COMPUTER PROFESSIONALS 1.	Good computer professional must be versatile 2.	Good computer professional must have ability to troubleshoot 3.	Good computer professional must have ability to carry out many jobs/assignments at the same time (Multi-tasking) 4.	Must be ready to improve himself 5.	Must be up to date on computer issues 6.	Must be able to solve computer problems 7.	Must have mathematical abilities. 8.	Must be diligent 9.	Must be ready to work with others a team 10.	Must be fluent in speaking 11.	Must be diligent

ICT AS A TRANSFORMATION TOOL IN OUR MODERN SOCIETY

ICT (Information and Communication Technology) refers to the technologies that provide access to information through telecommunication. The meaning of transformation Transformation deals with change. It could be from old to new stage of the societal demand. Transformation comes as a result of peoples’ desire and seeks for improvement in their activities. Information and Communication Technology (ICT), especially mobile phones, have revolutionized communications in Africa. The explosive growth of mobile phones in Africa over the past decade demonstrates the appetite for change across the continent. In the year 2000 there were fewer than 10 million fixed-line phones across Africa, a number that had accumulated slowly over a century, and a waiting list of a further 3.5 million. With a penetration rate of just over 1 per cent, phones were to be found only in offices and the richest households. But the coming of the mobile phone has transformed communications access. By the start of 2012, there were almost 650 million mobile subscriptions in Africa (A. T. Kearney, 2011), more than in the United States or the European Union2, making Africa the second fastest growing region in the world, after South Asia. At the start of the decade, few imagined that such demand existed, let alone that it could be afforded. In some African countries, more people have access to a mobile phone than to clean water, a bank account or even electricity. Mobile phones are now being used as a platform to provide access to the internet, to applications and to government services. The Role of ICT in Governments’ Activities Governments have an important role to play, in creating an enabling environment and in acting as a role model in adopting new innovations and technologies. Creating a vibrant environment where useful information is readily available to help entrepreneurs, farmers, health workers and environmentalists, for example, make better decisions in their daily activities requires a holistic approach and several supporting inputs or pillars. The key supporting pillars for such an environment includes adequate information and communications infrastructure, digital literacy and nurturing an ICT-skilled workforce that would propel emerging efforts to leverage ICTs to the next level to achieve sustainability and reliability. Taking a holistic view on a sector is a significant challenge for any government, regardless to how developed a country may be. Yet, as shown in the following chapters, African governments have made significant steps in building these pillars. In terms of infrastructure, much of Africa’s investments, private and public, have been in increasing network capacity or bandwidth so that the quality of internet or broadband service is available to more countries on the African continent. Infrastructure providing international connectivity requires large upfront investments which the private sector cannot shoulder. In these instances, public and donor funding are being leveraged. For example, in 2010 Eastern and Southern Africa was the only major region in the world not connected to the global broadband infrastructure by fibre optic cables. Twenty countries were reliant on expensive satellite connectivity to link with each other and the rest of the world. African governments and development financial institutions came together with the private sector to deploy the Eastern Africa Submarine Cable System, a submarine fibre-optic cable running 10,000 km along the east coast of Africa, connecting South Africa, Mozambique, Madagascar, Tanzania, Kenya, Somalia, Djibouti, Sudan, Comoros and Mayotte. Governments also participate directly in infrastructure investment, as the government of Botswana did when creating an alternative fibre route to the coast via Namibia. Hence, most of the international connectivity issues are being addressed. However, in order for ICT services to be accessible to more Africans, connectivity within the continent needs to be further improved. And the government’s larger role lies in creating an enabling environment – issuing licences, making available rights of way, managing spectrum, mandating infrastructure sharing and interconnection and so on – that allows a liberalized market to thrive and bring down price of service for the African consumer.

ICT AS A TRANSFORMATION TOOL IN OUR MODERN SOCIETY ICT (Information and Communication Technology) refers to the technologies that provide access to information through telecommunication. The meaning of transformation Transformation deals with change. It could be from old to new stage of the societal demand. Transformation comes as a result of peoples’ desire and seeks for improvement in their activities. Information and Communication Technology (ICT), especially mobile phones, have revolutionized communications in Africa. The explosive growth of mobile phones in Africa over the past decade demonstrates the appetite for change across the continent. In the year 2000 there were fewer than 10 million fixed-line phones across Africa, a number that had accumulated slowly over a century, and a waiting list of a further 3.5 million. With a penetration rate of just over 1 per cent, phones were to be found only in offices and the richest households. But the coming of the mobile phone has transformed communications access. By the start of 2012, there were almost 650 million mobile subscriptions in Africa (A. T. Kearney, 2011), more than in the United States or the European Union2, making Africa the second fastest growing region in the world, after South Asia. At the start of the decade, few imagined that such demand existed, let alone that it could be afforded. In some African countries, more people have access to a mobile phone

than to clean water, a bank account or even electricity. Mobile phones are now being used as a platform to provide access to the internet, to applications and to government services. The Role of ICT in Governments’ Activities Governments have an important role to play, in creating an enabling environment and in acting as a role model in adopting new innovations and technologies. Creating a vibrant environment where useful information is readily available to help entrepreneurs, farmers, health workers and environmentalists, for example, make better decisions in their daily activities requires a holistic approach and several supporting inputs or pillars. The key supporting pillars for such an environment includes adequate information and communications infrastructure, digital literacy and nurturing an ICT-skilled workforce that would propel emerging efforts to leverage ICTs to the next level to achieve sustainability and reliability. Taking a holistic view on a sector is a significant challenge for any government, regardless to how developed a country may be. Yet, as shown in the following chapters, African governments have made significant steps in building these pillars. In terms of infrastructure, much of Africa’s investments, private and public, have been in increasing network capacity or bandwidth so that the quality of internet or broadband service is available to more countries on the African continent.

Infrastructure providing international connectivity requires large upfront investments which the private sector cannot shoulder. In these instances, public and donor funding are being leveraged. For example, in 2010 Eastern and Southern Africa was the only major region in the world not connected to the global broadband infrastructure by fibre optic cables. Twenty countries were reliant on expensive satellite connectivity to link with each other and the rest of the world. African governments and development financial institutions came together with the private sector to deploy the Eastern Africa Submarine Cable System, a submarine fibre-optic cable running 10,000 km along the east coast of Africa, connecting South Africa, Mozambique, Madagascar, Tanzania, Kenya, Somalia, Djibouti, Sudan, Comoros and Mayotte. Governments also participate directly in infrastructure investment, as the government of Botswana did when creating an alternative fibre route to the coast via Namibia. Hence, most of the international connectivity issues are being addressed. However, in order for ICT services to be accessible to more Africans, connectivity within the continent needs to be further improved. And the government’s larger role lies in creating an enabling environment – issuing licences, making available rights of way, managing spectrum, mandating infrastructure sharing and interconnection and so on – that allows a liberalized market to thrive and bring down price of service for the African consumers.

GADGETS 1.	  MOBILE PHONE: A mobile phone is an electronic handheld device used to make mobile telephone calls comm.

INFORMATION AND COMMUNICATION TECHNOLOGY (ICT)-BASED GADGETS 2.	ICT (information and Communication Technology) involves the use of electronic computers and other electronic communication means to manage and process information effectively. 3.	ICT-based gadgets are basically devices used for information and communication technology. 4.	EXAMPLES OF ICTon manufactures of mobile phones are MOTOROLAR, SAMSUNG, BLACKBERRY, e t c 5.	COMPUTERS: A computer is an electronic device for storing and processing data. Computer come in various sizes like Person computer, Laptops, Ipad etc 6.	FAX MACHINE: fax stand for facsimile i.e. make a copy. It is a device that can send or receive pictures and texts over a telephone line. 7.	AUTOMATED TELLER MACHINE: An automatic teller machine (ATM), commonly called cash point is a computerized device that provides the clients of financial institutions access to financial transaction in a public space without the need of a cashier, human clerk or bank teller. 8.	   DISPENSING MACHINE (A VENDING MACHINE): This is a machine which dispenses items such as snacks, beverages, lottery, tickets, e t c. to customers automatically. 9.	 POINT OF SALE MACHINE /AUTOMATED CASH REGISTER (ACR): This is a machine that is used to carry out retail transactions. It can provide many services such as credit card processing, cash transaction e t c. 10. RADIO SETS: This is an electronic device that receives its input from an antenna, uses electronic fitters to separate a wanted radio signals from all other signals picked by this antenna, amplifies it to a level suitable for further processing and conveys through demodulation and decoding the signal into a form usable for consumer such as sound. 11.	TELEVISION SETS: A television set also called a TV .It is a device that is used to view television broadcast. Modern television consists of a display, antenna or radio frequency input and a tuner. 12.	DIGITAL CAMERA: This is used to capture image/picture in digital form. 13.	 PROJECTOR: This is an output device that can take images generated by a computer. It can be used to display information in a general public especially during public functions, teachings and seminars 14.	INTERNET: This is computer network that links computers together and allows almost all computers worldwide to connect and exchange information. 15.	SATELLITE: This is a complex device that is located in space that orbits (goes around) the earth. Satellite transmission is a method of information or signal carriage from one part of the earth to another. The method is used for television, telephone and other network transmission. Example, satellite transmission used by the Nigerian Television Authority. 16.	 TELEX MACHINE: This is a device that is used to send messages from one business to another on the telephone network or by satellite. 17.	PAGER: it is a device that connects into telephone line thereby transmitting a periodic signal as an indication that the conversation over the circuit is being recorded. 18.	BIOMETRIC SCANNER: It is used to check/capture fingerprint. 19.	REMOTE CONTROL: This i used to control information Technology devices such as volume control, channel control, on and off control etc. 20. FLASH DRIVE: For storage of information.

BUSINESS

Insurance 1. The objective of insurance is to reinstate the financial position of the insured to his or her previous position. 2. Taken to prevent risk or provide against it. 3. Insurance is based on the principle of indemnity. 4. The tenure of insurance is generally less. 5. The premium amount which is received is not the investment in other investment avenues to generate bonus.

Assurance 1. The objective of assurance is to pay the sum assured when the event takes place Taken 2. Taken against an event, whose occurrence is certain 3. Assurance is based on the principle of certainty 4. The tenure of assurance is more 5. The premium received by the assurance company is invested in other financial instruments to generate investment bonus will, in turn, increases the value of the policy The new topic today is : THEME: BOOK KEEPING AND BUSINESS SUCCESS TOPIC: LEDGER ENTRIES WHAT IS LEDGER ENTRIES? A ledger is the main book of accounts of any business. it is a large book which is divided into various units – each representing an account. all the transaction in a journal must be posted to the ledger. The ledger is very important because it is the basis for the double entry principle.

THEME: BOOK KEEPING AND BUSINESS SUCCESS TOPIC: LEDGER ENTRIES WHAT IS LEDGER ENTRIES? A ledger is the main book of accounts of any business. it is a large book which is divided into various units – each representing an account. all the transaction in a journal must be posted to the ledger. The ledger is very important because it is the basis for the double entry principle. Classes of ledger: 1.	Private ledger which consist of the account or records of transactions involving the owner of the business. It is made up of capital and drawings account. 2.	Personal ledger contains the accounts of debtors and creditors. The debtors are the people that have bought products from a business and have not paid yet. Creditor on the other hand are people from whom we have bought goods without making immediate payment. It is made up of credit sale and credit purchase account. 3.	General ledger comprises of accounts of assets, liabilities, incomes and expenses of a particular business. It consist of norminal and real accounts. Format of a ledger: Dr                                                              Ledger book                                                       Cr Date	Particular	Folio	Amount	Date	Particular	Folio	amount Items on a ledger: the ledger has two main sides, namely debit side(denote Dr) and credit side (denote Cr). The left hand is refer to as he debit side, while the right hand side is refers to as credit sides. The two sides may look the same but they perform different functions. Column 1 (Date) contains the records of the specific days on which transactions were carried out Column 2 (Particulars) is used for recording a summary of various transactions Column 3 (Folio) commonly refers to ledger folio is the column in which the page numbers of a specific account are entered. Column 4 (Amount) this column is used to record the value of each transaction. Note that a transaction could be settled using cash and cheques. ASSIGNMENT: 1.	List two accounts found in a private ledger and four account found in general ledger 2.	Describe the four columns found on each side of the ledger Cashbook You will recall that journals are books of original entry, while the ledger is the principal book of account.

The cashbook is a book in which all cash transactions are recorded. It is a special book in accounting. Note that credit transactions are not recorded in the cashbook.

Classes of cashbook

There are four main categories of cashbook. They are:

The single column cashbook: this is concerned only with cash receipts and cash payments

The double column cashbook: the double column cashbook records cash and cheque receipt and payments. It has an additional column for bank.

The three column cashbook covers cash and cheque receipts and payments as well as cash discounts given and received. Hence, it has extra column for cash, bank and discount.

The petty cashbook deals with minor expenses or payments. This unit will consider only the first two classes or categories.

CRS

Teacher Mr. Victor Alabi Date: 16/06/2020. Class: Basic 8 Subject: C.R.S Topic :Christian As Salt And Light Of The World( cont)

Matthew 5: 13-16 - Christians as Salt of the world(cont) If Christians refused to performed their duties as Christians, then he or she is like salt which has lost its taste- he or she can not be reclaimed( how shall its saltness be restored?) impossible; he is useless and unprofitable( it is no longer good for anything); and he or she is doomed to ruin and rejection( to be thrown out and trodden under foot by men). Therefore, as a child of God or Christian try to possess good characters and let these good characters preach the gospel to unbeliever. - Christian as Light Of The World. Light is the ray that shown all over places to see hidden things clearly. Christians are also the light of the world. " Your are the light of the world" (Mathew 5:14) The following are some qualities of light. i.	it is always welcome. ii. ii. it drives away darkness ii. it shows the true nature of somethings.etc Therefore, Christians are expected : iii. . to lead others to Christ iv. . to drives away the darkness of ignorance and fear. etc read, copy the note a.nd ask me questions

1. Salt is a mineral composed primarily of sodium chloride (NaCl), a chemical compound belonging to the larger class of salts; salt in its natural form as a crystalline mineral is known as rock salt or halite. Salt is present in vast quantities in seawater, where it is the main mineral constituent. 2. Chemically, salt is sodium chloride. It has a vitreous luster, and its color usually ranges from colorless to white, but occasionally it is red, yellow or blue. Among its notable features: it is highly diathermic, plastic, viscous and flows at high pressures. 3. The general theme of Matthew 5:13–16 is promises and expectations, and these expectations follow the promises of the first part. The first verse of this passage introduces the phrase "salt of the earth": You are the salt of the earth, but if the salt has lost its flavor, with what will it be salted? Assignment. a. find out and state four more qualities Christians must have as salt of the earth. b. find out and explain the meaning of light. c. find out and state four qualities Christians must have as light of the world.

Teacher : Victor Alabi Date: 23/06/2020 Class: Basic 8 Subject: C.R.S Topic: Christians as Light of the world (cont) - Christians as light (cont). iii. Christians as light of the earth are expected to show others an example of how God wants them to live by their own good actions and holy lives. Moreover, as the light of the world, Christians are important in the society and have an eye on us. Some admire us, commend us etc, while others envy us, hate us etc. But when a Christian makes cannot hide because a lot of people are watching them. “A city set on a hill cannot be hidden." All our neighbours up his or her mind to live in the way Christ has ordered. He or she is an example of how to live and behave, and he has equally become the light of the earth. Therefore, as a child of God, before you can be salt and light of the world, you must put all. Jesus' teaching into practice in your own daily life.

Teacher : Victor Alabi Date : 14/07/ 2020 Class : basic 8 Subject : C.R.S Topic : Sermon on the mount- Forgiveness (Matt 6 : 14-15 ) – Definition of forgiveness Forgiveness is pardoning an offender for whatever crime or offence he or she has committed no matter how serious the offence may be. No doubt that everyone who is alive today has sinned and thereby needs forgiveness. We sin against each other and against God. Therefore, we can forgive each other if we humbly beg for forgiveness. However, it is very important to forgive because Jesus taught that God will forgive you only if you forgive others( Matt 6: 14-15). – Conditions for forgiveness, are: a. repentance b. the willingness to make amends. – Importance of forgiveness, are: a. it restores broken relationship b. it maintain our relationship with God. c. it prevents circle of revenge etc. Teacher: Victor Alabi Date: 28/07/20 Class: Basic 8 Subject: C.R.S Topic: Sermon On The Mount: Fasting and Prayer. - Prayer ( Mathew 6: 5-13) *Definition of prayer. Prayer is the act of communicating to God for one's heart desire. Jesus taught on how to pray in Matt 6: 5-13. The teaching base on the following. i. He warn the disciples against hypocrisy- the desire to show off ii. The key to answer prayer is to do it in secret- go into your room and shut the door and God will hear and answer. iii. Prayer should not consist of vain repetitions- pilling up words upon words, God is not impressed by many words. iv. We pray because, in prayer we show that we agree that have needs and that only God can meet them for us or answer the prayer. v.	Finally, Jesus taught the sequences through which prayer must follow. This is called 'The Lord's Prayer'. 'Our Father Who Hath In Heaven, Hallow Be Thy Name, Thy Kingdom Come .etc.

Date : 11/08/20. Subject : CRS Topic : Sermon On The Mount- Fasting and Prayer ( cont ).

- The Importance Of Fasting. Fasting is very important in the life of a Christian as a result of the following reasons. i. it teaches us self- discipline ii. it gives us time to pray. iii. it helps us appreciate God's gifts iv. it reminds us that we can live without depends so much on food.

- Sermon On The Mount- Love of  Money( Mathew 6: 19-20) Money is important because it helps us to meet some of our  needs. When we love money too much to the extend of do anything to get, that when it becomes bad Therefore Jesus warns against strongly desiring of things of the world, especially money in Mathew 6:19-20. Note: read the teaching of Jesus on love of money from your Bible.

ECONOMICS

PERSONAL HYGIENE

PUBERTY: is a short span of time that marks the beginning of sexual maturation. This period differs between boys and girls. Boys generally reach puberty at about the age of 14years, while girls at about the age of 12years.

SIGNS OF PUBERTY IN BOYS 1. Hairs begin to grow on different parts of the body such as the armpit, pubic region, beard around the jaw and a moustache above the upper lip. 2. Voice breaks and becomes deeper 3. There is change in general body appearance as muscles develop. The boy may need larger and new clothes. 4. Sex organs develop. Sex glands called testicles produce spermatozoa or sperm.

SIGNS OF PUBERTY IN GIRLS 1. The breast develop 2. The body changes 3. Hair grows on her armpit and pubic region 4. Menstruation starts. 5. There could be skin changes and problem, such as pimples.

MENSTRUATION

Menstruation is the monthly flow of blood from the womb, through the vagina which occurs in every woman of child bearing age. This period is called menstrual period. it takes place after about every 25--30 days.

MENSTRUAL HYGIENE PRACTICE 1. Use good absorbent sanitary pads or towel and pants. These will prevent your clothes from being stained by blood. 2. Change the sanitary pads as often as three times a day to prevent bad odour. 3. Wrap the solid pad in an old news paper. 4. Have frequent baths during menstruation. At list bath twice a day. Have local bath of the vulva area each time you change your pad if possible. 5. Menstruation is not a disease. Therefore carry about your normal activities during your menstrual period. 6. Not using perfume to cover unpleasant odour 7. Applying perfume on clean bodies and clothes.

IMPORTANCE OF PERSONAL HYGIENE 1. It helps an individual to cope with such temporary physical conditions as oily skin, and hair, and increase perspiration. 2. It helps to remove body odour 3. It results in better health 4. It gives a more attractive appearance. 5. It gives an individual the confidence needed to be in the company of mates.

WAYS OF MAINTAINING GOOD PERSONAL HYGIENE 1. Daily brushing of the teeth 2. Maintaining clean hair, hands and nails. 3. Washing hands before eating.

CLASS WORK

1. The short span of time which marks the beginning of sexual maturation is called A. PUBERTY B. ADOLESCENCE C. CHILDHOOD D. ADULTHOID

2. The period which lie between the end of childhood and adulthood is called A. ADOLESCENCE B. BOYHOOD C. PUBERTY D. MANHOOD

3. The age of puberty in boys and girls is A. DIFFERENT B. THE SAME C. PROLONGED D. CERTAIN

4. Breast development in girls is A ALWAYS A SIGN OF PUBERTY B. A SIGN OF ILLNESS C. A SIGN OF CLASS D. NOT ALWAYS A SIGN OF PUBERTY

5. The unpleasant smell that comes from the body is called body A. Odour B. Perfume C. Smell D. Complex

MEANING OF CONFLICTS A Conflict is a struggle between two or more people who disagree. That struggle may be verbal, physical or both. A family conflict is a struggle between two or more family members who disagree over issues.

CAUSES OF CONFLICTS IN FAMILIES 1. SITUATIONAL CAUSES: situation that dissatisfies people can cause Conflict. For instance, in families, a. When parent or parents show more love and attention to a child and neglects the other. b. When a husband fails to gives his wife money for house keep. c. When children disobey their parents.

2. PERSONALITY DIFFERENCE: No two individual are the same, not even twins. Some people may be slow and quiet, while others could be fast, noisy and outgoing. These different behavioral patterns can bring about conflicts in the family.

3. POWER STRUGGLES: Conflicts can occur when people feel a need to be in control. Each person will be struggling for power. This can occur between husband and wife. It can also occur between a younger and elder brother or sister. In some cases, a boy might try to show a girl that he is stronger than she.

NEGATIVE RESULTS OF CONFLICTS Conflicts need to be handled or resolved properly. When they are not properly handled, negative results occur. 1. Negative emotions arise, such as anger, frustration, fear, pain, humiliation, sorrow or bitterness, etc.

2. People who are often angry may become I'll. Conflict causes stress. Stress is often linked to ulcers and heart diseases. 3. People may say things they do not mean in the heart of anger. 4. Relationship suffers. Conflicts can break up friendships and families. 5. Violence may occur. When temper arises with serious arguments, there can be physical attack. Injury or even death can result.

IMPORTANCE GUIDELINES IN CONFLICT RESOLUTION 1. Use words not fists 2. Take a decision to resolute the conflict peacefully. Take charge of the situation. 3. Try to talk in a place other people will not distract you or interfere with your efforts 4. When you talk, take turns. No one person should dominate the talking. Every person must be given a chance to talk. 5. Use active listening and keep an open mind. 6. Show respect to the other person, recur respect from him/her 7. Control your voice. 8. Speak the truth. 9. Control your tongue.

FAMILY CRISIS MEANING OF FAMILY CRISIS Family crisis is a situation that marks a turning point, when things cease to go on as usual in the family. Crisis can be challenges. They need to be managed properly. DIFFERENT TYPES OF FAMILY CRISIS 1.	ARRIVAL OF A NEW BORN BABY: The arrival of a new baby is normally a joy to the family but it can also be a  sources of crisis. The father may fell neglected as mother turns all her attention to the baby and other children may also fell neglected and become jealous. WAYS OF MANAGING THE CRISIS

A. The family should plan and prepare probably before each new baby arrives

2.	CLASHES OF PERSONALITY: Family members differ in their abilities, likes and dislike. They also differ in the way they react to situations. Some are fast while others are slow. However when they disagree seriously there may be clashes in personality. WAYS OF MANAGING THE CRISIS

A. Family should set Family values, life styles, goals and standard. These should be made known to each member, so that they can work together.

3.	RELOCATION OF FAMILY: sometimes a family may have to move from the community, town or state where they live to another. They have to get used to their new location. These situations are sources of crisis.

WAYS OF MANAGING THE CRISIS

A. have a positive attitude B. Give the new community and people a chance.

4. CHANGE OF JOB/EMPLOYMENT: Many people change jobs several times during their careers. Home maker/wife or mother who was not working before might get a job. Such job related Change can affect a family in a numbers of ways. For instance, it can bring changes in income that requires adjustment and decision about spending. WAYS OF MANAGING THE CRISIS A. Family goals will need to be reviewed and Family members may have to take up responsibilities. 5. DIVORCE: means the break up of a marriage. This is one of the most serious crisis in any family. It is even more serious when children are involved and parents disagree over custody. CUSTODY is the legal responsibility of housing and caring for children. Divorce has a negative effect on husband and wife, children and in-laws. WAYS OF MANAGING THE CRISIS A. parents need to agree on how to help the children B. Children should be allowed to communicate with parents C. Counseling can help. 6. PROBLEMS AT SCHOOL: some children have different problems at schools. Such problems include: examination malpractices, failure in examination, participating in cult activities, suspension and truancy. These create crises in families. WAYS OF MANAGING THE CRISIS A. Parents should show understanding towards children. B. The causes of the problems must be identified and solutions to the problems must be sought. C. There should be open communication among parents, children and school.