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Splash Screen
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Lesson Menu
Five-Minute Check (over Chapter 5) Then/Now New
Vocabulary Theorem 6.1 Polygon Interior Angles
Sum Example 1 Find the Interior Angles Sum of a
Polygon Example 2 Real-World Example Interior
Angle Measure of Regular Polygon Example 3 Find
Number of Sides Given Interior Angle
Measure Theorem 6.2 Polygon Exterior Angles
Sum Example 4 Find Exterior Angle Measures of a
Polygon
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5-Minute Check 1
State whether this sentence is always, sometimes,
or never true. The three altitudes of a triangle
intersect at a point inside the triangle.
A. always B. sometimes C. never

• A
• B
• C

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5-Minute Check 2
Find n and list the sides of ?PQR in order from
shortest to longest if m?P 12n 15, m?Q 7n
26, and m?R 8n 47.

1. A
2. B
3. C
4. D

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5-Minute Check 3
State the assumption you would make to start an
indirect proof of the statement. If 2x 18,
then x 9.
A. x is positive. B. 2x lt 18 C. x gt 9 D. x lt 9

1. A
2. B
3. C
4. D

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5-Minute Check 4
Find the range for the measure of the third side
of a triangle given that the measures of two
sides are 43 and 29.
A. n gt 14 B. 14 lt n lt 72 C. 29 lt n lt 43 D. n gt 0

1. A
2. B
3. C
4. D

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5-Minute Check 5
Write an inequality relating m?ABD and m?CBD.
A. m?ABD lt m?CBD B. m?ABD m?CBD C. m?ABD gt
m?CBD D. m?ABD m?CBD

1. A
2. B
3. C
4. D

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5-Minute Check 6
Write an equation that you can use to find the
measures of the angles of the triangle.
A. x 5 3x 180 B. x 5 3x 111
180 C. x 5 3x 69 D. x 5 3x 111

1. A
2. B
3. C
4. D

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Then/Now
You named and classified polygons. (Lesson 16)

• Find and use the sum of the measures of the
  interior angles of a polygon.

• Find and use the sum of the measures of the
  exterior angles of a polygon.

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Vocabulary

• diagonal

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Concept 1
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Example 1A
Find the Interior Angles Sum of a Polygon
A. Find the sum of the measures of the interior
angles of a convex nonagon.
A nonagon has nine sides. Use the Polygon
Interior Angles Sum Theorem to find the sum of
its interior angle measures.
(n 2) ? 180 (9 2) ? 180 n 9 7 ? 180
or 1260 Simplify.
Answer The sum of the measures is 1260.
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Example 1B
Find the Interior Angles Sum of a Polygon
B. Find the measure of each interior angle of
parallelogram RSTU.
Step 1 Find x.
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Example 1B
Find the Interior Angles Sum of a Polygon
Sum of measures of interior angles
Substitution
Combine like terms.
Subtract 8 from each side.
Divide each side by 32.
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Example 1B
Find the Interior Angles Sum of a Polygon
Step 2 Use the value of x to find the measure of
each angle.
m ? R 5x 5(11) or 55
m ? S 11x 4 11(11) 4 or 125
m ? T 5x 5(11) or 55
m ? U 11x 4 11(11) 4 or 125
Answer m?R 55, m?S 125, m?T 55, m?U 125
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Example 1A
A. Find the sum of the measures of the interior
angles of a convex octagon.
A. 900 B. 1080 C. 1260 D. 1440

1. A
2. B
3. C
4. D

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Example 1B
B. Find the value of x.
A. x 7.8 B. x 22.2 C. x 15 D. x 10

1. A
2. B
3. C
4. D

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Example 2
Interior Angle Measure of Regular Polygon
ARCHITECTURE A mall is designed so that five
walkways meet at a food court that is in the
shape of a regular pentagon. Find the measure of
one of the interior angles of the pentagon.
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Example 2
Interior Angle Measure of Regular Polygon
Solve Find the sum of the interior angle
measures. (n 2) ? 180 (5 2) ? 180 n
5 3 ? 180 or 540 Simplify. Find the measure
of one interior angle.
Substitution
Divide.
Answer The measure of one of the interior angles
of the food court is 108.
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Example 2
A pottery mold makes bowls that are in the shape
of a regular heptagon. Find the measure of one of
the interior angles of the bowl.
A. 130 B. 128.57 C. 140 D. 125.5

1. A
2. B
3. C
4. D

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Example 3
Find Number of Sides Given Interior Angle Measure
The measure of an interior angle of a regular
polygon is 150. Find the number of sides in the
polygon.
Use the Interior Angle Sum Theorem to write an
equation to solve for n, the number of sides.
S 180(n 2) Interior Angle Sum Theorem (1
50)n 180(n 2) S 150n 150n 180n
360 Distributive Property 0 30n 360 Subtract
150n from each side.
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Example 3
Find Number of Sides Given Interior Angle Measure
360 30n Add 360 to each side. 12 n Divide
each side by 30.
Answer The polygon has 12 sides.
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Example 3
The measure of an interior angle of a regular
polygon is 144. Find the number of sides in the
polygon.
A. 12 B. 9 C. 11 D. 10

1. A
2. B
3. C
4. D

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Concept 2
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Example 4A
Find Exterior Angle Measures of a Polygon
A. Find the value of x in the diagram.
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Example 4A
Find Exterior Angle Measures of a Polygon
Use the Polygon Exterior Angles Sum Theorem to
write an equation. Then solve for x.
5x (4x 6) (5x 5) (4x 3) (6x 12)
(2x 3) (5x 5) 360 (5x 4x 5x 4x
6x 2x 5x) (6) (5) 3 (12) 3
5 360 31x 12 360 31x 372 x 12
Answer x 12
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Example 4B
Find Exterior Angle Measures of a Polygon
B. Find the measure of each exterior angle of a
regular decagon.
A regular decagon has 10 congruent sides and 10
congruent angles. The exterior angles are also
congruent, since angles supplementary to
congruent angles are congruent. Let n the
measure of each exterior angle and write and
solve an equation. 10n 360 Polygon Exterior
Angle Sum Theorem n 36 Divide each side by
10.
Answer The measure of each exterior angle of
a regular decagon is 36.
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Example 4A
A. Find the value of x in the diagram.
A. 10 B. 12 C. 14 D. 15

1. A
2. B
3. C
4. D

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Example 4B
B. Find the measure of each exterior angle of a
regular pentagon.
A. 72 B. 60 C. 45 D. 90

1. A
2. B
3. C
4. D

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End of the Lesson