3.4 Find Angle Measures in Polygons Thompson Objectives: 1. To find the sum of the measures of the interior and exterior angles in any n-gon.

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Presentation transcript:

3.4 Find Angle Measures in Polygons Thompson Objectives: 1. To find the sum of the measures of the interior and exterior angles in any n-gon

Example 1 What is the sum of the interior angles in the polygon below?

Example 2 What’s the difference between convex and concave polygons?

Project Review Using the two previous concepts, we will discover a method for finding the sum of the angles in any convex n -gon, where n is the number of sides (or angles) of a given polygon. Step 1: Draw a series of convex n -gons, starting with n = 4 and ending with n = 8.

Project Review Step 2: In each polygon, draw all of the diagonals from one vertex. Notice how these diagonals divide the polygons into triangles. How could this help find the sum of the angles in each n -gon?

Project Review Step 3: Complete the table. Number of sides 3456 Number of triangles Angle Sum

Project Review Step 4: Find a formula. Number of sides 3456 Number of triangles 1234 Angle Sum180°360°540°720°

Polygon Interior Angles Theorem The sum of the measures of the interior angles of a convex n -gon is ( n – 2)·180°. m  1 + m  2 + … + m  n = ( n – 2)·180°

Example 3 What is the sum of the measures of the interior angles of a convex octagon?

Example 4 What is the measure of each angle of an equiangular octagon?

Example 5 Find the values of e and f.

Example 6 What is the measure of each angle in any equiangular n -gon?

Equiangular Polygon Conjecture The measure of each angle of an equiangular n -gon can be found by using either of the following expressions:

Example 7 In a regular polygon, the measure of each angle is 150 . How many sides does the polygon have?

Example 8: From SAT If the degree measures of the angles of a quadrilateral are 4 x, 7 x, 9 x, and 10 x, what is the sum of the measures of the smallest angle and the largest angle?

Project Review When you extend one side of a triangle, you form an exterior angle. If you extend each side of a polygon to form one exterior angle at each vertex, you create a set of exterior angles for the polygon.

Polygon Exterior Angles Theorem The sum of the measures of one set of exterior angles of a polygon is 360°.

Example 9 What is the value of x ?

Example 10 What is the number of sides of a polygon in which the sum of the degree measures of the interior angles is 4 times the sum of the degree measures of the exterior angles?

Example 11 What is the measure of each exterior angle in an equiangular octagon? What is the measure of each exterior angle in an equiangular n -gon? How does this relate to the Equiangular Polygon Conjecture?