Find and use the measures of interior and exterior angles of polygons.

Objectives

6.1: Polygon Angle Theorems

Handbook p. 19

Polygon Angle Sum Theorem:

The sum of the interior angles of a convex polygon is (n 2) 180.Draw each diagonal from a single vertex, then complete the chart.

Polygon # of Sides # of Triangles Sum of Interior Measures

Triangle

Quadrilateral

Pentagon

Hexagon

n-gon

6.1: Polygon Angle Theorems

1. Find the sum of the interior angle measures of a convex heptagon.

(n – 2)180°

(7 – 2)180°

900°

2. Find the measure of each interior angle of a regular 16-gon.

Step 1: Find the sum of the interior angle measures.

Step 2: Find the measure of one interior angle.

(n – 2)180°(16 – 2)180° = 2520°

6.1: Polygon Angle Theorems

Interior Angles Exterior Angles

Sum of all angles

Measure of one if the polygon is REGULAR!

)2(180 n

n

n )2(180

Handbook page 19

3. Find the measure of each interior angle of pentagon ABCDE.

(5 – 2)180° = 540°

35c + 18c + 32c + 32c + 18c = 540

135c = 540c = 4

6.1: Polygon Angle Theorems

mA = 35(4°) = 140°

mB = mE = 18(4°) = 72°

mC = mD = 32(4°) = 128°

4a. What is the measure of each interior angle of a regular decagon?

b. What is the measure of each exterior angle of a regular decagon?

c. What is the SUM of all of the exterior angles?

6.1: Polygon Angle Theorems

In the polygons below, an exterior angle has been measured at each vertex. Notice that in each case, the

sum of the exterior angle measures is 360°.

6.1: Polygon Angle Theorems

5. Find the measure of each exterior angle of a regular 20-gon.

A 20-gon has 20 sides and 20 vertices.

Sum of ext. s = 360°.

The measure of each exterior angle of a regular 20-gon is 18°.

Measure of one ext. =

Angle measures of a convex polygon with n sides

Interior Angles Exterior Angles

Sum of all angles

Measure of one if the polygon is REGULAR!

)2(180 n

n

n )2(180

360

n

360

6. Find the value of b in polygon FGHJKL.

15b° + 18b° + 33b° + 16b° + 10b° + 28b° = 360°

120b = 360

b = 3

7. Name the convex polygon whose sum of its interior angle measures is 6840.

8. The exterior angle of a regular polygon is about 27.7. Find the number of its sides and the measure of each interior angle.