User talk:Cespher

1.GEOMETRIC RELATION -	angles are often classified by their relationship to other angles or to other parts of a geometric figure. For example, angles 1 and 3 in

A.Relations involving segment

B.Angles and Side of a Triangle By relative lengths of sides

Triangles can be classified according to the relative lengths of their sides: •Equilateral triangle, all sides are of equal length. An equilateral triangle is also an equiangular polygon, i.e. all its internal angles are equal—namely, 60°; it is a regular polygon. •Isosceles triangle, two sides are of equal length (originally and conventionally limited to exactly two). An isosceles triangle also has two equal angles: the angles opposite the two equal sides. •Scalene triangle, all sides have different lengths. The internal angles in a scalene triangle are all different.

By internal angles Triangles can also be classified according to their internal angles, described below using degrees of arc: •Right triangle (or right-angled triangle, formerly called a rectangled triangle) has one 90° internal angle (a right angle). The side opposite to the right angle is the hypotenuse; it is the longest side in the right triangle. The other two sides are the legs or catheti (singular: cathetus) of the triangle. Right triangles conform to the Pythagorean theorem, wherein the sum of the squares of the two legs is equal to the square of the hypotenuse, i.e., a2 + b2 = c2, where a and b are the legs and c is the hypotenuse. See also Special right triangles •Oblique triangle has no internal angle equal to 90°. •Obtuse triangle is an oblique triangle with one internal angle larger than 90° (an obtuse angle). •Acute triangle is an oblique triangle with internal angles all smaller than 90° (three acute angles). An equilateral triangle is an acute triangle, but not all acute triangles are equilateral triangles.

C.	Angles form by parallel lines cut by a transversal

2.TRIANGLE CONGRUENTS - two pairs of corresponding angles are congruent (therefore the third pair of corresponding angles are also congruent). -the three pairs of corresponding sides are proportional.

A. Condition for Triangle Congruence A few basic postulates and theorems about similar triangles: •Two triangles are similar if at least two corresponding angles are equal. •If two corresponding sides of two triangles are in proportion, and their included angles are congruent, then triangles are similar. •If three sides of two triangles are in proportion, then the triangles are similar. (Definition of similar triangles)

For two triangles to be congruent, each of their corresponding angles and sides must be equal (6 total). A few basic postulates and theorems about congruent triangles: •SAS Postulate - If two sides and the included angle (angle between the two sides) of two triangles are congruent, then the triangles are congruent. •ASA Postulate - If two angles and the included side (side between the two angles) of two triangles are congruent, then the triangles are congruent. •SSS Postulate - If all three sides of two triangles are congruent, then the triangles are congruent. •AAS Theorem - If two angles and a non-included side of two triangles are congruent, then the triangles are congruent. •Hypotenuse-Leg (HL) Theorem: If a leg and hypotenuse of two right triangles are congruent, then the triangles are congruent (if the triangle is not a right triangle, then it'd be the SSA condition, which does not guarantee congruent triangles). •Hypotenuse-Angle Theorem: If the hypotenuse and an acute angle of one right triangle are congruent to a hypotenuse and an acute angle of another right triangle, then the triangles are congruent •Side-Side-Angle (or Angle-Side-Side) condition: if two sides and an angle that isn't included of two triangles are equal, then if the angle is obtuse, the opposite side is longer than the adjacent, or the opposite side is equal to the sine of the angle times the adjacent side, the triangles are congruent. ***The SSA condition does NOT guarantee congruent triangles.

B. Using Triangle Congruent

3.QUADRILATERAL -is a polygon with four sides. The parts of a quadrilateral are its sides, its four angles, and its two DIAGONALS. A diagonal is a straight line joining two alter AC and DB are the diagonals

A.The Quadrilateral -is a polygon with four sides or edges and four vertices or corners.

Convex quadrilaterals are further classified as follows: •Trapezoid: two opposite sides are parallel. •Isosceles trapezoid: two opposite sides are parallel and the base angles are congruent. This implies that the other two sides are of equal length, and that the diagonals are of equal length. An alternative definition is a quadrilateral with an axis of symmetry bisecting one pair of opposite sides. •Trapezium: no sides are parallel. (In British English this would be called an irregular quadrilateral, and was once called a trapezoid.) •Parallelogram: both pairs of opposite sides are parallel. This implies that opposite sides are of equal length, opposite angles are equal, and the diagonals bisect each other. A general term including square, rectangle, rhombus and rhomboid. •Kite: two adjacent sides are of equal length and the other two sides also of equal length. This implies that one set of opposite angles is equal, and that one diagonal perpendicularly bisects the other. (It is common, especially in the discussions on plane tessellations, to refer to a concave kite as a dart or arrowhead.) •Rhombus or rhomb: all four sides are of equal length, or congruent. This implies that opposite sides are parallel, opposite angles are equal, and the diagonals perpendicularly bisect each other. "A pushed-over square." •Rhomboid: a parallelogram in which adjacent sides are of unequal lengths and angles are oblique (not right angles). "A pushed-over rectangle." •	Rectangle (or Oblong): all four angles are congruent right angles. This implies that opposite sides are parallel and of equal length, and the diagonals bisect each other and are equal in length, or congruent. •Square (regular quadrilateral): all four sides are of equal length (equilateral), and all four angles are congruent, with each angle a right angle. This implies that opposite sides are parallel (a square is a parallelogram), and that the diagonals perpendicularly bisect each other and are of equal length. A quadrilateral is a square if and only if it is both a rhombus and a rectangle. •Rhombus (four congruent sides) + Rectangle (four congruent angles) = Square (four congruent sides and four congruent angles) --> Parallelogram (opposite sides are parallel) --> Quadrilateral (four-sided polygon) •Cyclic quadrilateral: the four vertices lie on a circumscribed circle. •Tangential quadrilateral: the four edges are tangential to an inscribed circle. Another term for a tangential polygon is inscriptible. •Bicentric quadrilateral: both cyclic and tangential.

4.SIMILARITY -two geometrical objects are called similar if one is congruent to the result of a uniform scaling (enlarging or shrinking) of the other.

A.Ration and Proportion

Ratio is a comparison of two numbers. We generally separate the two numbers in the ratio with a colon.

Proportion is an equation with a ratio on each side. It is a statement that two ratios are equal.

B.Similarity between triangles Similar triangles: If triangle ABC is similar to triangle DEF, then this relation can be denoted as

In order for two triangles to be similar, it is sufficient for them to have at least two angles that match. If 	this is true, then the third angle will also match, since the three angles of a triangle must add up to 180°.

Suppose that triangle ABC is similar to triangle DEF in such a way that the angle at vertex A is congruent 	with the angle at vertex D, the angle at B is congruent with the angle at E, and the angle at C is congruent 	with the angle at F. Then, once this is known, it is possible to deduce proportionalities between 	corresponding sides of the two triangles, such as the following: In summary, If two triangles are similar, all the linear objects in the two triangles are of the same ratio.

This idea can be extended to similar polygons with any number of sides. That is, given any two similar polygons, the corresponding sides are proportional.

Angle/side similarities: A concept commonly taught in high school mathematics is that of proving the "angle" and "side" 	theorems, which can be used to define two triangles as similar (or indeed, congruent). In each of these three-letter acronyms, A stands for equal angles, and S for equal sides. For example, ASA refers to an angle, side and angle that are all equal and adjacent, in that order. •AAA - Angle-Angle-Angle. If two triangles share three common angles, they are similar. (Obviously, this means that the side lengths are locked in a common ratio, but can vary proportionally, making the triangles similar.) Additionally, since the interior angles of a triangle have a sum of 180°, having two triangles with only two common angles (sometimes known as AA) implies similarity as well.

C.Proportionality Given two variables x and y, y is (directly) proportional to x if there is a non-zero constant k such that the relation is often denoted and the constant ratio is called the proportionality constant or constant of proportionality.

Properties: Since it follows that if y is proportional to x, with (nonzero)proportionality constantthen x is also proportional to y with proportionality constant 1/k.is equivalent to y is proportional to x, then the graph of y as a function of x will be a straight line passing through the origin with the slope of the line equal to the constant of proportionality. —Preceding unsigned comment added by 119.95.26.45 (talk) 09:47, 10 January 2009 (UTC)