d You found the relationship in Part C based on a pattern.
E
e The sums of angle measures are multiples of 180.
A
a See solution.
B
b See solution.
C
cExample Solution: When the number of sides is increased by 1, the sum of the angle measures is increased by 180.
D
d Inductive, see solution.
E
e (n-2)180
Practice makes perfect
a Let's draw the polygons and measure the angles as asked.
Make sure you measure the angles accurately. You will probably draw different polygons and your angle measures will be different.
b Let's copy the angle measures from the figures of Part A to the table. In the last column we will write the sum of the measures in the corresponding rows.
Number of Sides
Angle Measures
Sum of Angle Measures
3
m∠ A
64
m∠ C
69
180
m∠ B
47
4
m∠ F
81
m∠ H
67
360
m∠ G
127
m∠ J
85
5
m∠ P
93
m∠ S
104
540
m∠ Q
139
m∠ T
101
m∠ R
103
c Let's look at the differences between the sums of the angle measures.
180 +180 ⟶ 360 +180 ⟶ 540
We can see that as the number of sides of the polygon is increased by 1, the sum of the angle measures is increased by 180.
e Notice that the sums of the angle measures are all multiples of 180.
Number of Sides
Sum of Angle Measures
3
180= 1* 180
4
360= 2* 180
5
540= 3* 180
We can see that the multiplier of 180 is 2 less than the number of sides.
This observation allows us to write the expression for the sum of the measures of the angles for a polygon with n sides.
( n-2)180
