Five-Minute Check (over Chapter 5)

CCSS

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

Over Chapter 5

A. x – 5 + 3x = 180

B. x – 5 + 3x + 111 = 180

C. x – 5 + 3x = 69

D. x – 5 + 3x = 111

Write an equation that you can use to find the measures of the angles of the triangle.

Over Chapter 5

A. x – 5 + 3x = 180

B. x – 5 + 3x + 111 = 180

C. x – 5 + 3x = 69

D. x – 5 + 3x = 111

Write an equation that you can use to find the measures of the angles of the triangle.

Content Standards

G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

Mathematical Practices

4 Model with mathematics.

3 Construct viable arguments and critique the reasoning of others.

You named and classified polygons.

• 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.

• diagonal

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:

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.

Find the Interior Angles Sum of a Polygon

B. Find the measure of each interior angle of parallelogram RSTU.

Since the sum of the measures of the interior angles is Write an equation to express the sum of the measures of the interior angles

of the polygon.

Step 1 Find x.

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.

Find the Interior Angles Sum of a Polygon

Step 2 Use the value of x to find the measure of each angle.

Answer:

mR = 5x= 5(11) or 55

mS = 11x + 4= 11(11) + 4 or 125

mT = 5x= 5(11) or 55

mU = 11x + 4= 11(11) + 4 or 125

Find the Interior Angles Sum of a Polygon

Step 2 Use the value of x to find the measure of each angle.

Answer: mR = 55, mS = 125, mT = 55, mU = 125

mR = 5x= 5(11) or 55

mS = 11x + 4= 11(11) + 4 or 125

mT = 5x= 5(11) or 55

mU = 11x + 4= 11(11) + 4 or 125

A. 900

B. 1080

C. 1260

D. 1440

A. Find the sum of the measures of the interior angles of a convex octagon.

A. 900

B. 1080

C. 1260

D. 1440

A. Find the sum of the measures of the interior angles of a convex octagon.

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.

Interior Angle Measure of Regular Polygon

Solve Find the sum of the interiorangle measures.

(n – 2) ● 180 = (5 – 2) ● 180 n = 5

= 3 ● 180 or 540Simplify.

Find the measure of one interiorangle.

Substitution

Divide.

Interior Angle Measure of Regular Polygon

Answer: The measure of one of the interior angles of the food court is 108.

Check To verify that this measure is correct,

use a ruler and a protractor to draw a

regular pentagon using 108 as themeasure of each interior angle. Thelast side drawn should connect with

thebeginning point of the first segmentdrawn.

Find Exterior Angle Measures of a Polygon

A. Find the value of x in the diagram.

Find Exterior Angle Measures of a Polygon

Use the Polygon Exterior Angles Sum Theorem to write an equation. Then solve for x.

Answer:

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

Find Exterior Angle Measures of a Polygon

Use the Polygon Exterior Angles Sum Theorem to write an equation. Then solve for x.

Answer: x = 12

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

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 AngleSum Theorem

n = 36 Divide each side by 10.

Answer:

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 AngleSum Theorem

n = 36 Divide each side by 10.

Answer: The measure of each exterior angle of aregular decagon is 36.

A. 10

B. 12

C. 14

D. 15

A. Find the value of x in the diagram.

A. 10

B. 12

C. 14

D. 15

A. Find the value of x in the diagram.

A. 72

B. 60

C. 45

D. 90

B. Find the measure of each exterior angle of a regular pentagon.

A. 72

B. 60

C. 45

D. 90

B. Find the measure of each exterior angle of a regular pentagon.