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24. Angles of a Polygon

Competency 23 : Makes decisions regarding day to day activities based on geometrical concepts related to rectilinear plane figures.

Competency Level 23.3 : Performs calculations using the sum of the exterior and interior angles of polygons.

Time : 80 minutes.

Learning –Teaching Process:

Step 1 : Present the following set of figures to the class and lead a discussion on the shape of each plane figure and the angles in the figures.

During this discussion, highlight the following facts.

Step 2 : Engage the students in an analytical study by using the following leaflet on exploration. • That a closed plane figure bounded by straight line segments is defined as a polygon. • That a closed plane figure bounded by three sides is defined as a triangle and that the triangle is the polygon with the least number of sides. • That a closed plane figure bounded by four sides is defined as a quadrilateral. • That the sum of the interior angles of a triangle equals 180o. • That the sum of the interior angles of a quadrilateral equals 360o. • That the exterior angles of a polygon are obtained by producing its sides. • That the number of interior angles as well as the number of exterior angles of a polygon equals the number of sides of the polygon.

(20 minutes)

Leaflet on Exploration

• Focus your attention on the polygon received by your group from the following polygons.

• Determine the number of sides the polygon you received has and propose a name for it. • Divide the polygon into triangles using a common vertex and calculate the sum of the interior angles of the polygon by using the result on the sum of the interior angles of a triangle. • Draw another polygon with as many sides as you like and in the same manner as above, calculate the sum of the interior angles. • By considering a triangle, a quadrilateral and the polygon you received, determine whether there is a relationship between the number of sides of a polygon and the sum of its interior angles. • On another piece of paper make an enlarged copy of the polygon you received and draw all its exterior angles. • Cut out all the exterior angles and paste them in a manner such that they do not overlap but such that their vertices coincide. • What can you say about the sum of the exterior angles? • Draw another polygon with as many sides as you like and determine whether the result you obtained above can be confirmed. • What can you say about the sum of an interior angle of a polygon and the related exterior angle? • If the sides of the polygon you received were of equal length and if all the interior angles too were of equal magnitude, propose a suitable name for the polygon. • Prepare to present your group’s findings at the plenary session.

Step 3 :

After the students’ presentations, lead a discussion and highlight the following facts.

Criteria for Assessment and Evaluation: • Calculates the sum of the interior angles, the sum of the exterior angles of a polygon when the number of sides of the polygon is given. • When the magnitude of some of the interior or exterior angles of a polygon is given, calculates the magnitude of the other angles. • For a regular polygon, determines the magnitude of an angle when the number of sides is known, and the number of sides when the magnitude of an angle is known. • Applies generalized results to special situations. • Works in cooperation within the group.

• That depending on the number of sides it has, a polygon can be classified as a quadrilateral, a pentagon, a hexagon etc. • That if any polygon is divided into triangles, all having a common vertex which is also a vertex of the polygon, then the number of triangles obtained is 2 less that the number of sides of the polygon. • That the sum of the interior angles of any polygon can be found by considering the number of triangles the polygon can be divided into, and using the result on the sum of the interior angles of a triangle. • That the sum of the exterior angles of any polygon is 360o. • That the sum of the exterior angles of a polygon is independent of the number of sides of the polygon. • That for a polygon, the magnitude of an interior angle + the magnitude of the corresponding exterior angle = 180o. • That a polygon with sides of equal length and interior angles of equal magnitude is defined as a regular polygon. • That when the magnitude of an exterior angle of a regular polygon is known, the number of sides of the polygon can be found by using the relationship

• That when the number of sides of a regular polygon is known, the magnitude of an exterior angle can be found by using the relationship

(30 minutes)