Assignment• P. 510-513: 1, 2-10

even, 11-21, 24-28, 37-41

• Inscribed Polygons Worksheet

Example 1What is the sum of the interior angles in the

polygon below?

S

U

M

Example 2What’s the difference between convex and

concave polygons?

Investigation 1Using 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 = 3 and ending with n = 6.

Investigation 1

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

Investigation 1

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

Investigation 1

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°

8.1 Find Angle Measures in Polygons

Objectives:1. To find the sum of the measures of the

interior and exterior angles in any n-gon

Polygon Interior Angles TheoremThe sum of the measures of the interior

angles of a convex n-gon is (n – 2)·180°.

m1 + m2 + … + mn = (n – 2)·180°

Example 3What is the sum of the measures of the

interior angles of a convex octagon?

Example 4What is the measure of each angle of an

equiangular octagon?

Example 5Find the values of e and f.

Example 6What is the measure of each angle in any

equiangular n-gon?

Equiangular Polygon TheoremThe measure of each angle of an

equiangular n-gon can be found by using either of the following expressions:

nnn

360180or 180)2(

Example 7In a regular polygon, the measure of each

angle is 150˚. How many sides does the polygon have?

Example 8: SATIf 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?

Investigation 2When 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.

Investigation 2Use the GSP activity

to investigate the sum of the measures of a set of exterior angles of a polygon.

Polygon Exterior Angles TheoremThe sum of the measures of one set of

exterior angles of a polygon is 360°.

Example 9What is the value of x?

Example 10What 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 11What 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 Theorem?

Assignment• P. 510-513: 1, 2-10

even, 11-21, 24-28, 37-41

• Inscribed Polygons Worksheet