8.1 Find Angle Measures in 8.1 Find Angle Measures in PolygonsPolygons

Objectives:

1. To find the sum of the measures of the interior and exterior angles in any n-gon

Example 1Example 1

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

S

U

M

Example 2Example 2

What’s the difference between convex and concave polygons?

Investigation 1Investigation 1

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 = 3 and ending with n = 6.

Investigation 1Investigation 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 1Investigation 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 1Investigation 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°

Polygon Interior Angles Polygon Interior Angles TheoremTheoremThe sum of the measures of the interior

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

m1 + m2 + … + mn = (n – 2)·180°

Example 3Example 3

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

Example 4Example 4

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

Example 5Example 5

Find the values of e and f.

Example 6Example 6

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

Equiangular Polygon ConjectureEquiangular 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 7Example 7

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

Example 8: SATExample 8: 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?

Investigation 2Investigation 2

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.

Investigation 2Investigation 2

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

Polygon Exterior Angles Polygon Exterior Angles TheoremTheorem

The sum of the measures of one set of exterior angles of a polygon is 360°.

Example 9Example 9

What is the value of x?

Example 10Example 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 11Example 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?

AssignmentAssignment

• P. 510-513: 1, 2-10 even, 11-21, 24-28, 37-41

• Inscribed Polygons Worksheet