2.7

What is Its Measure?

Pg. 23

Interior Angles of a Polygon

2.7 – What is Its Measure?_____________Interior Angles of a Polygon

In an earlier chapter, you discovered that the sum of the interior angles of a triangle is always 180°. But what about the sum of the interior angles of other polygons, such as hexagons or decagons? Does it matter if the polygon is convex or not? Consider these questions today as you investigate the angles of a polygon.

2.36 – INTERIOR ANGLESWith your team, find the sum of the measures of its interior angles as many ways as you can. You may want to use the fact that the sum of the angles of a triangle is 180°. Be prepared to share your method with the class.

72°72°

72°72°72°

54° 54°

54° 54°

54°

54°

54° 54°

54°

54°

540°

a

b

c

d

e

f

g

h i

180° 180°

180°

540°

2.37 – SUM OF THE INTERIOR ANGLES OF A POLYGONIn the previous problem, you found the sum of the measures of the interior angles of a pentagon. But what about other polygons?

a. Use your method from the previous problem to find the sum of the interior angles of other polygons. Draw a sketch to help support your answers. Find the sum of the measures of several shapes. (It might be helpful to split up the tasks within your group).

b. Complete the chart below based on your findings.

# of sides of polygon

3 4 5 6

Sum of interior angles

180° 360° 540° 720°

b. Complete the chart below based on your findings.

# of sides of polygon

7 8 9 10

Sum of interior angles

1080° 1260° 1440°900°

b. Complete the chart below based on your findings.

# of sides of polygon

15 25 50

Sum of interior angles

4140° 8640°2340°

c. What if it was has "n" number of sides? What is the sum of the interior angles then? Come up with a formula to find the sum of the interior angles of any polygon.

180(n – 2)

where n = # of sides

2.38 – FINDING ANGLESUse the angle relationships in each of the diagrams below to solve for the given variables. Show all work.

12

3

45

6

7

900°

3m + 501 =3m = 399m = 133°

180(n – 2)180(7 – 2)

900

6x =x = 120°

720°

180(n – 2)180(6 – 2)

720

1080°

32x + 824 =32x = 256

x = 8°

180(n – 2)180(8 – 2)

1080

2.39 – USING INTERIOR AND EXTERIOR ANGLESUse your understanding of polygons to answer the questions below, if possible. If there is no solution, explain why not.

a. How many sides does a polygon have if the sum of the measures of the interior angles is 1980°? 2,520°?

a. How many sides does a polygon have if the sum of the measures of the interior angles is 1980°? 2,520°?

180(n – 2) = 1980n – 2 = 11

n = 13

n – 2 = 14n = 16

180(n – 2) = 2520

b. A quadrilateral has four sides. What is the measure of each of its interior angles?

Depends on if it is regular or not

180(n – 2) n

=

180(n – 2) = 162n180n – 360 = 162n

–360 = –18n

20 = n

180(n – 2) n

each interior angle =

c. What if you know that the measure of an interior angle of a regular polygon is 162°? How many sides must the polygon have?

162°1

c. Each interior angle of a regular pentagon has measure 2x + 4°. What is x?

Sum of pentagon:

180(5 – 2) =540°

Each angle is 2x + 4, so all 5 add to….

5(2x + 4) = 540

10x + 20 = 54010x = 520

x = 52°

2.40 – CENTRAL VS INTERIOR ANGLESBelow is a list of regular polygons. Find the measure of each central angle and each interior angle. Show all work.

360/8 = 45°

45°1080°

180(n – 2)180(8 – 2)

1080/8 = 135°

135°

360/5 = 72°

72° 540°

180(n – 2)180(5 – 2)

540/5 = 108°

108°

360/40 = 9°

6840°

180(n – 2)180(40 – 2)

6840/5 = 171°

2.41 – CONCLUSIONSHow do you find the sum of the interior angles? How is this measure different than the central angle?

Sum of the Interior Angles

Each interior angle in a regular polygon

180(n – 2) 180(n – 2)n