3.5- Part 2 Angle Measures in

Polygons

Investigation: Sum of the interior angles

1. Draw examples of 4-sided, 5-sided, and 6-sided polygons. In each polygon, draw all the diagonals from 1 vertex.

2. Complete the table on the next slide. What is the pattern in the sum of the measures of the interior angles in any convex n-gon?

Polygon # of Sides # of triangles

Sum of angles

Triangle 3 1 180°

Quad

Pentagon

Hexagon

n-gon n

Polygon # of Sides # of triangles

Sum of angles

Triangle 3 1 180°

Quad 4 2 2·180=360°

Pentagon 5 3 3·180=540°

Hexagon 6 4 4·180=720°

Dodecagon 12 10 10·180= 1800

n-gon n n-2 (n-2)180°

Polygon Interior Angles Theorem

The sum of the interior angles of a convex n-gon is

(n-2)•180°.One angle in a regular n-gon:

€

n − 2( ) •180nBM #34-35

Exterior Angles

Polygon Exterior Angle Theorem

The sum of the measures of the exterior angles of a

convex n-gon is 360°.

BM #36

1

2

3

4

5

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n • interior +exterior =180n°

interior = n − 2( )180° or 180n − 360°

So, 180n − 360 + exterior = 180n−360 + exterior = 0exterior = 360°

Example 1:

• A heptagon has 4 interior angles that measure 160° each and two interior angles that are right angles. What is the measure of the other interior angle?

BM #34

Ex. 1 Solution:

• (n-2)180=interior sum

• (7-2)180=5•180=900°

• 4•160+2•90=640+180=820°

• 900-820=80°

Example 2:

• Find the measure of each angle in a regular 11-gon.

BM #35

Ex. 2 solution:

• (n-2)180=(11-2)180

• 9•180=1620°

• 1620÷11=147.3°

Example 3:

• The measure of each exterior angle of a regular polygon is 40°. How many sides does the polygon have?

BM #36

Ex. 3 Solution:

• 360÷40=9

• 9 sides

Assignment:

#22 Polygon Worksheet

#23 3.5 WS (p. 301)