FIND ANGLE MEASURES IN
POLYGONS
Ch 8.1
Vocab

Consecutive vertices: vertices that are next to each
other or part of the same side.
Diagonal

A diagonal is a segment that connects 2 nonconsecutive edges.
Polygon Interior Angles Theorem

Theorem 8.1
What is the sum of the interior angles
of…
(n  2) 180
(3  2) 180  180

A triangle:

A quadrilateral:

A pentagon:
(5  2) 180  540

A hexagon:
(6  2) 180  720

A heptagon:
(4  2) 180  360
(7  2) 180  900
Corollary to Interior Angle Theorem
If I have the Sum of the interior angles how
do I figure out how many sides there are?
(n  2) 180  sum of angles
Given that the sum is 1800
(n  2) 180  1800
(n  2)  180 1800

180
180
n  2  10
n  12
If I have the Sum of the interior angles how
do I figure out how many sides there are?
(n  2) 180  sum of angles
Given that the sum is 1980
(n  2) 180  1980
(n  2)  180 1980

180
180
n  2  11
n  13
How would you find x?
What kind of figure is this?
What is the sum of the interior
angles?
108  121  59  x  360
288  x  360
x  72
What kind of figure is this?
What is the sum of the interior
angles of a pentagon?
(5 – 2)(180) = 540
110+92+100+84 + x = 540
386 + x = 540
x = 154
What kind of figure is it?
How do you find x?
What is the sum of the interior
angles of a heptagon?
(7 – 2)(180) = 900
145+112+99+133+156+2x+x = 900
645 + 3x = 900
3x = 255
x = 85
Finding exterior angles
How do you find x?
67 + 96 + 59 + 86 + x =
360
308 + x = 360
x = 52
Find x
X = 71
Page 510 #3 - 16
Word Problems

What is the measure of one exterior angle in a regular
triangle?
120 ̊
 Recall that the sum of any convex
polygon’s exterior angles is 360.
 Then divide by how many exterior
angles there are in the shape.
 360 ÷ 3 = 120
What is the measure of an exterior
angle in a regular 10-gon?

360 ÷ 10 = 36 ̊
The measures of the exterior angles of
a convex pentagon are 54, 72, 2x, 3x
and x. What is the measure of the
largest angle?






The sum of the measure of the exterior angles
equals 360, so we’ll add the angles and find x.
54 + 72 + 2x + 3x + x = 360
126 + 6x = 360
6x = 234
X = 39
3x = 3(39) = 117
Find an interior and exterior angle for
a regular 11-gon

360 ÷ 11 = 32.73 ̊

X + 32.73 = 180

X = 147.27
How do you find how many sides a figure
has, when given one angle measure?



How do you find the sum of the interior angles?
(n – 2)180 = interior angle sum
If we have a regular figure all the angles are the
same, so we would divide the interior angle sum by
the number of sides, n, to find the measure of one
angle. (n  2)180
n
 one angle measure
An interior angle of a regular polygon has
a measure of 150 ̊. How many sides does
the figure have?(n  2)180
n
 one angle measure
(n  2)180
 150 Multiply both sides by n!
n
(n  2)180
n
 150  n
n
(n  2)180  150 n Distribute the 180!
180 n  360  150 n Subtract the 180n!
 360  30 n
12  n
An interior angle of a regular polygon has
a measure of 120 ̊. How many sides does
the figure have?(n  2)180
n
 one angle measure
(n  2)180
 120 Multiply both sides by n!
n
(n  2)180
n
 120  n
n
(n  2)180  120n Distribute the 180!
180 n  360  120 n Subtract the 180n!
 360  60 n
6n
Find the measure of and interior and exterior angle of a
regular hexagon.
Find the measure of and interior and exterior angle of a
regular 20-gon.
A regular polygon has an angle measure of 108 ̊ how
many sides does the polygon have?