3.4 Find Angle Measures in PolygonsThompson

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?

S

U

M

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?

180

180

180180

180

180

180

180

180180

Project Review

Step 3: Complete the table.

180

180

180180

180

180

180

180

180180

Number of sides

3 4 5 6

Number of triangles

Angle Sum

Project Review

Step 4: Find a formula.

180

180

180180

180

180

180

180

180180

Number of sides

3 4 5 6

Number of triangles

1 2 3 4

Angle Sum 180° 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°.

m1 + m2 + … + mn = (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 a regular octagon?

Example 5

Find the values of e and f.

Example 6

What is the measure of each angle in any regular 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:

nn

n

360180or

180)2(

Example 7

In a regular n-gon, 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 4x, 7x, 9x, and 10x, 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 a regular octagon?

What is the measure of each exterior angle in an regular n-gon?

Classwork

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