5.7 Angle Measures in
Polygons
Vocabulary/Theorems
 Diagonal:
joins 2 nonconsecutive
vertices
 Convex Polygon: has no vertex
going into the interior of the
polygon
 Concave
Polygon: has a vertex
going into the interior of the
polygon
 Sum
of the measures of the
interior angles of a convex
polygon is 180(n-2)o
 Sum
of the measures to the
exterior angles of a convex
polygon is 360o
 In
a regular polygon, the
interior angles are congruent.
EXAMPLE 1
Find the sum of angle measures in a polygon
Find the sum of the measures
of the interior angles of a
convex octagon.
SOLUTION
An octagon has 8 sides. Use the Polygon Interior
Angles Theorem.
Substitute 8 for n.
(n – 2) 180° = (8 – 2) 180°
Subtract.
= 6 180°
= 1080°
Multiply.
ANSWER
The sum of the measures of the interior angles of an
octagon is 1080°.
EXAMPLE 2
Find the number of sides of a polygon
The sum of the measures of the interior angles of a
convex polygon is 900°. Classify the polygon by the
number of sides.
SOLUTION
Use the Polygon Interior Angles Theorem to write an
equation involving the number of sides n. Then solve
the equation to find the number of sides.
Polygon Interior Angles Theorem
(n –2) 180° = 900°
Divide each side by 180°.
n –2 = 5
n =7
Add 2 to each side.
ANSWER
The polygon has 7 sides. It is a heptagon.
GUIDED PRACTICE
for Examples 1 and 2
1. The coin shown is in the
shape of a regular 11- gon.
Find the sum of the measures
of the interior angles.
ANSWER 1620°
2. The sum of the measures of the interior angles of
a convex polygon is 1440°. Classify the polygon
by the number of sides.
ANSWER
decagon
With a partner, do #1-2 on p. 299
EXAMPLE 3
Find an unknown interior angle measure
ALGEBRA Find the value of x in the diagram shown.
SOLUTION
The polygon is a
quadrilateral. Write an
equation involving x.
Then solve the equation.
x° + 108° + 121° + 59° = 360°
x + 288 = 360
x = 72
ANSWER
Corollary to Theorem 8.1
Combine like terms.
Subtract 288 from each side.
The value of x is 72.
GUIDED PRACTICE
for Example 3
3. Use the diagram at the right.
Find m S and m T.
ANSWER
103°, 103°
4. The measures of three of the interior angles of a
quadrilateral are 89°, 110°, and 46°. Find the
measure of the fourth interior angle.
ANSWER
115°
EXAMPLE 4
Standardized Test Practice
SOLUTION
x° + 2x° + 89° + 67° = 360° Polygon Exterior Angles Theorem
3x + 156 = 360 Combine like terms.
ANSWER
Solve for x.
x = 68
The correct answer is B.
GUIDED PRACTICE
for Example 4
5. A convex hexagon has exterior angles with
measures 34°, 49°, 58°, 67°, and 75°. What is the
measure of an exterior angle at the sixth vertex?
ANSWER
77°
Do #3, 4 on p. 299