Download Find Angle Measures in Polygons

8.1
Find Angle Measures in
Polygons
Goal
Your Notes
p Find angle measures in polygons.
VOCABULARY
Diagonal
THEOREM 8.1: POLYGON INTERIOR ANGLES
THEOREM
The sum of the measures of the
interior angles of a convex n-gon
is (n 2
)p
.
m∠1 1 m∠2 1 . . . 1 m∠n 5 (n 2
2
1
6
3
5
4
n56
)p
COROLLARY TO THEOREM 8.1: INTERIOR ANGLES OF
A QUADRILATERAL
The sum of the measures of the interior angles of a
quadrilateral is
.
Example 1
Find the sum of angle measures in a polygon
Find the sum of the measures of the
interior angles of a convex hexagon.
Solution
A hexagon has
Theorem.
(n 2
)p
sides. Use the Polygon Interior Angles
5(
5
5
2
p
)p
Substitute
for n.
Subtract.
Multiply.
The sum of the measures of the interior angles of a
hexagon is
.
206
Lesson 8.1 • Geometry Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
8.1
Find Angle Measures in
Polygons
Goal
Your Notes
p Find angle measures in polygons.
VOCABULARY
Diagonal A diagonal of a polygon is a segment that
joins two nonconsecutive vertices.
THEOREM 8.1: POLYGON INTERIOR ANGLES
THEOREM
2
The sum of the measures of the
3
1
interior angles of a convex n-gon
6
5
is (n 2 2 ) p 1808 .
m∠1 1 m∠2 1 . . . 1 m∠n 5 (n 2 2 ) p 1808
4
n56
COROLLARY TO THEOREM 8.1: INTERIOR ANGLES OF
A QUADRILATERAL
The sum of the measures of the interior angles of a
quadrilateral is 3608 .
Example 1
Find the sum of angle measures in a polygon
Find the sum of the measures of the
interior angles of a convex hexagon.
Solution
A hexagon has 6 sides. Use the Polygon Interior Angles
Theorem.
(n 2 2 ) p 1808 5 ( 6 2 2 ) p 1808
Substitute
6 for n.
5 4 p 1808
Subtract.
5 7208
Multiply.
The sum of the measures of the interior angles of a
hexagon is 7208 .
206
Lesson 8.1 • Geometry Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
Find the number of sides of a polygon
Example 2
The sum of the measures of the interior angles of a
convex polygon is 12608. 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.
(n 2
)p
n2
5
Polygon Interior Angles
Theorem
5
Divide each side by
n5
The polygon has
Example 3
Add
.
to each side.
sides. It is a
.
Find an unknown interior angle measure
Find the value of x in the diagram shown.
x8
718
Solution
The polygon is a quadrilateral. Use the
1358 1128
Corollary to the Polygon Interior Angles Theorem
to write an equation involving x. Then solve the equation.
x8 1
1
1
5
Corollary to
Theorem 8.1
x1
5
Combine like
terms.
x5
Subtract
from each side.
Checkpoint Complete the following exercise.
1. Find the sum of the measures
of the interior angles of the
convex decagon.
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Lesson 8.1 • Geometry Notetaking Guide
207
Your Notes
Example 2
Find the number of sides of a polygon
The sum of the measures of the interior angles of a
convex polygon is 12608. 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.
(n 2 2 ) p 1808 5 12608
Polygon Interior Angles
Theorem
n2 2 5 7
Divide each side by 1808 .
n5 9
Add 2 to each side.
The polygon has 9 sides. It is a nonagon .
Example 3
Find an unknown interior angle measure
Find the value of x in the diagram shown.
x8
718
Solution
The polygon is a quadrilateral. Use the
1358 1128
Corollary to the Polygon Interior Angles Theorem
to write an equation involving x. Then solve the equation.
x8 1 1358 1 1128 1 718 5 3608
x 1 318 5 360
x 5 42
Corollary to
Theorem 8.1
Combine like
terms.
Subtract 318
from each side.
Checkpoint Complete the following exercise.
1. Find the sum of the measures
of the interior angles of the
convex decagon.
14408
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Lesson 8.1 • Geometry Notetaking Guide
207
Your Notes
Checkpoint Complete the following exercises.
2. The sum of the measures of the interior angles of a
convex polygon is 16208. Classify the polygon by the
number of sides.
3. Use the diagram at the right.
Find m∠K and m∠L.
N
1058
J
688
1098 M
K
L
THEOREM 8.2: POLYGON EXTERIOR ANGLES
THEOREM
4
1
5
n55
Find unknown exterior angle measures
Example 4
Find the value of x in the diagram shown.
Solution
Use the Polygon Exterior Angles Theorem
to write and solve an equation.
x8 1
1
Lesson 8.1 • Geometry Notetaking Guide
898
x8
858
2x8
1
5
Polygon Exterior
Angles Theorem
x1
5
Combine like
terms.
x5
208
3
2
The sum of the measures of the exterior
angles of a convex polygon, one angle at
.
each vertex, is
m∠1 1 m∠2 1 . . . 1 m∠n 5
Solve for x.
Copyright © Holt McDougal. All rights reserved.
Your Notes
Checkpoint Complete the following exercises.
2. The sum of the measures of the interior angles of a
convex polygon is 16208. Classify the polygon by the
number of sides.
11-gon
3. Use the diagram at the right.
Find m∠K and m∠L.
m∠K 5 m∠L 5 1298
N
1058
J
688
1098 M
K
L
THEOREM 8.2: POLYGON EXTERIOR ANGLES
THEOREM
Example 4
4
1
5
n55
Find unknown exterior angle measures
Find the value of x in the diagram shown.
Solution
Use the Polygon Exterior Angles Theorem
to write and solve an equation.
x8 1 2x8 1 858 1 898 5 3608
3 x 1 174 5 360
x 5 62
208
Lesson 8.1 • Geometry Notetaking Guide
3
2
The sum of the measures of the exterior
angles of a convex polygon, one angle at
each vertex, is 3608 .
m∠1 1 m∠2 1 . . . 1 m∠n 5 3608
898
x8
858
2x8
Polygon Exterior
Angles Theorem
Combine like
terms.
Solve for x.
Copyright © Holt McDougal. All rights reserved.
Your Notes
Example 5
Find angle measures in regular polygons
Lamps The base of a lamp is in the shape of a regular
15-gon. Find (a) the measure of each interior angle and
(b) the measure of each exterior angle.
Solution
a. Use the Polygon Interior Angles Theorem to find the
sum of the measures of the interior angles.
(n 2
)p
5(
2
)p
5
Then find the measure of one interior angle. A regular
15-gon has
congruent interior angles.
Divide
by
:
4
5
.
The measure of each interior angle in the
15-gon is
.
b. By the Polygon Exterior Angles Theorem, the sum of
the measures of the exterior angles, one angle at each
vertex, is
. Divide
by
:
4
5
.
The measure of each exterior angle in the 15-gon
is
.
Checkpoint Complete the following exercises.
4. A convex pentagon has exterior angles with
measures 668, 778, 828, and 628. What is the
measure of an exterior angle at the fifth vertex?
Homework
5. Find the measure of (a) each interior angle and
(b) each exterior angle of a regular nonagon.
Copyright © Holt McDougal. All rights reserved.
Lesson 8.1 • Geometry Notetaking Guide
209
Your Notes
Example 5
Find angle measures in regular polygons
Lamps The base of a lamp is in the shape of a regular
15-gon. Find (a) the measure of each interior angle and
(b) the measure of each exterior angle.
Solution
a. Use the Polygon Interior Angles Theorem to find the
sum of the measures of the interior angles.
(n 2 2 ) p 1808 5 ( 15 2 2 ) p 1808
5 23408
Then find the measure of one interior angle. A regular
15-gon has 15 congruent interior angles.
Divide 23408 by 15 : 23408 4 15 5 1568 .
The measure of each interior angle in the
15-gon is 1568 .
b. By the Polygon Exterior Angles Theorem, the sum of
the measures of the exterior angles, one angle at each
vertex, is 3608 . Divide 3608 by 15 :
3608 4 15 5 248 .
The measure of each exterior angle in the 15-gon
is 248 .
Checkpoint Complete the following exercises.
4. A convex pentagon has exterior angles with
measures 668, 778, 828, and 628. What is the
measure of an exterior angle at the fifth vertex?
738
Homework
5. Find the measure of (a) each interior angle and
(b) each exterior angle of a regular nonagon.
a. 1408
b. 408
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Lesson 8.1 • Geometry Notetaking Guide
209