CorrectionKey=NL-A;CA-A CorrectionKey=NL-D;CA-D ......2018/02/23 · interior angle of the...

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Explore 1 Exploring Interior Angles in TrianglesYou can find a relationship between the measures of the three angles of a triangle. An interior angle is an angle formed by two sides of a polygon with a common vertex. So, a triangle has three interior angles.

Use a straightedge to draw a large triangle on a sheet of paper and cut it out. Tear off the three corners and rearrange the angles so their sides are adjacent and their vertices meet at a point.

What seems to be true about placing the three interior angles of a triangle together?

Make a conjecture about the sum of the measures of the interior angles of a triangle.

The conjecture about the sum of the interior angles of a triangle can be proven so it can be stated as a theorem. In the proof, you will add an auxiliary line to the triangle figure. An auxiliary line is a line that is added to a figure to aid in a proof.

The Triangle Sum Theorem

The sum of the angle measures of a triangle is 180°.

Fill in the blanks to complete the proof of the Triangle Sum Theorem.

Given: △ABC

Prove: m∠1 + m∠2 + m∠3 = 180°

Statements Reasons

1. Draw line ℓ through point B parallel to _ AC . 1. Parallel Postulate

2. m∠1 = m∠ and m∠3 = m∠ 2.

3. m∠4 + m∠2 + m∠5 = 180° 3. Angle Addition Postulate and definition of

straight angle

4. m∠ + m∠2 + m∠ = 180° 4.

Resource Locker

interior angles

A C

B ℓ

1

2

3

4 5

They form a straight angle.

The sum of the measures of the interior

angles of a triangle is 180°.

4 5 Alternate Interior Angles Theorem

Substitution Property of Equality31

Module 7 313 Lesson 1

7 . 1 Interior and Exterior AnglesEssential Question: What can you say about the interior and exterior angles of a triangle

and other polygons?

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Common Core Math StandardsThe student is expected to:

G-CO.C.10

Prove theorems about triangles.

Mathematical Practices

MP.8 Patterns

Language ObjectiveWork in small groups to play angle jeopardy.

COMMONCORE

COMMONCORE

HARDCOVER PAGES 271282

Turn to these pages to find this lesson in the hardcover student edition.

Interior and Exterior Angles

ENGAGE Essential Question: What can you say about the interior and exterior angles of a triangle and other polygons?The sum of the interior angle measures of a triangle

is 180°. You can find the sum of the interior angle

measures of any n-gon, where n represents the

number of sides of the polygon, by multiplying

(n - 2) 180°. In a polygon, an exterior angle forms a

linear pair with its adjacent interior angle, so the

sum of their measures is 180°. In a triangle, the

measure of an exterior angle is equal to the sum of

the measures of its two remote interior angles.

PREVIEW: LESSON PERFORMANCE TASKView the Engage section online. Discuss the photo, asking students to recall and describe the designs of game boards of their favorite games. Then preview the Lesson Performance Task.

313

HARDCOVER

Turn to these pages to find this lesson in the hardcover student edition.

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Explore 1 Exploring Interior Angles in Triangles

You can find a relationship between the measures of the three angles of a triangle.

An interior angle is an angle formed by two sides of a polygon with a common vertex.

So, a triangle has three interior angles.

Use a straightedge to draw a large triangle on a sheet

of paper and cut it out. Tear off the three corners and

rearrange the angles so their sides are adjacent and

their vertices meet at a point.

What seems to be true about placing the three

interior angles of a triangle together?

Make a conjecture about the sum of the measures

of the interior angles of a triangle.

The conjecture about the sum of the interior angles of a triangle can be proven so it can be stated

as a theorem. In the proof, you will add an auxiliary line to the triangle figure. An auxiliary line

is a line that is added to a figure to aid in a proof.

The Triangle Sum Theorem

The sum of the angle measures of a triangle is 180°.

Fill in the blanks to complete the proof of the Triangle Sum Theorem.

Given: △ABC

Prove: m∠1 + m∠2 + m∠3 = 180°

Statements

Reasons

1. Draw line ℓ through point B parallel to _

AC . 1. Parallel Postulate

2. m∠1 = m∠ and m∠3 = m∠ 2.

3. m∠4 + m∠2 + m∠5 = 180°

3. Angle Addition Postulate and definition of

straight angle

4. m∠ + m∠2 + m∠ = 180° 4.

Resource

Locker

G-CO.C.10 Prove theorems about triangles.

COMMONCORE

interior angles

A

C

B ℓ

1

2

3

4 5

They form a straight angle.

The sum of the measures of the interior

angles of a triangle is 180°.

45 Alternate Interior Angles Theorem

Substitution Property of Equality

31

Module 7

313

Lesson 1

7 . 1 Interior and Exterior Angles

Essential Question: What can you say about the interior and exterior angles of a triangle

and other polygons?

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313 Lesson 7 . 1

L E S S O N 7 . 1

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Reflect

1. Explain how the Parallel Postulate allows you to add the auxiliary line into the triangle figure.

2. What does the Triangle Sum Theorem indicate about the angles of a triangle that has three angles of equal measure? How do you know?

Explore 2 Exploring Interior Angles in PolygonsTo determine the sum of the interior angles for any polygon, you can use what you know about the Triangle Sum Theorem by considering how many triangles there are in other polygons. For example, by drawing the diagonal from a vertex of a quadrilateral, you can form two triangles. Since each triangle has an angle sum of 180°, the quadrilateral must have an angle sum of 180° + 180° = 360°.

A Draw the diagonals from any one vertex for each polygon. Then state the number of triangles that are formed. The first two have already been completed.

B For each polygon, identify the number of sides and triangles, and determine the angle sums. Then complete the chart. The first two have already been done for you.

PolygonNumber of

SidesNumber of Triangles

Sum of Interior Angle Measures

Triangle 3 1 (1)180° = 180°

Quadrilateral 4 2 (2)180° = 360°

Pentagon ( ) 180° =

Hexagon ( ) 180° =

Decagon ( ) 180° =

quadrilateral

2 triangles

quadrilateraltriangle

1 triangle 2 triangles3 triangles

4 triangles 5 triangles 6 triangles

quadrilateraltriangle

1 triangle 2 triangles3 triangles

4 triangles 5 triangles 6 triangles

Since there is only one line parallel to a given line that passes through a given point, I can

draw that line into the triangle and know it is the only one possible.

180 _ 3

= 60, so each angle of the triangle must have a measure of 60°.

5 3

6 4

10 8

3 540°

720°

1440°

4

8


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Integrate Mathematical PracticesThis lesson provides an opportunity to address Mathematical Practice MP.8, which calls for students to “look for and identify patterns.” Throughout the lesson, students use hands-on investigations or geometry to predict patterns and relationships for the interior and exterior angles of a triangle or polygon. They prove the Triangle Sum Theorem, the Polygon Angle Sum Theorem, and the Exterior Angle Theorem. The hands-on investigations give students a chance to use inductive reasoning to make a conjecture. This is followed by a proof in which students use deductive reasoning to justify their conjectures.

EXPLORE 1 Exploring Interior Angles in Triangles

INTEGRATE TECHNOLOGYStudents have the option of completing the interior angles in triangles activity either in the book or online.

QUESTIONING STRATEGIESWhat can you say about angles that come together to form a straight line? Why? The

sum of the angle measures must be 180° by the

definition of a straight angle and the Angle Addition

Postulate.

Is it possible for a triangle to have two obtuse angles? Why or why not? No; the sum of

these angles would be greater than 180°.

EXPLORE 2 Exploring Interior Angles in Polygons

AVOID COMMON ERRORSWhen attempting to determine the sum of the interior angles of a polygon, some students may divide the figure into too many triangles. For example, a student may draw both diagonals of a quadrilateral and conclude that the sum of the interior angles of a polygon is 720°. Point out that four of the angles of the triangles are not part of an interior angle of the quadrilateral, and demonstrate the correct division.

PROFESSIONAL DEVELOPMENT

Interior and Exterior Angles 314


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Do you notice a pattern between the number of sides and the number of triangles? If n represents the number of sides for any polygon, how can you represent the number of triangles?

Make a conjecture for a rule that would give the sum of the interior angles for any n-gon.

Sum of interior angle measures =

Reflect

3. In a regular hexagon, how could you use the sum of the interior angles to determine the measure of each interior angle?

4. How might you determine the number of sides for a polygon whose interior angle sum is 3240°?

[# Explain 1 Using Interior AnglesYou can use the angle sum to determine the unknown measure of an angle of a polygon when you know the measures of the other angles.

Polygon Angle Sum Theorem

The sum of the measures of the interior angles of a convex polygon with n sides is (n – 2) 180°.

Example 1 Determine the unknown angle measures.

For the nonagon shown, find the unknown angle measure x°.

First, use the Polygon Angle Sum Theorem to find the sum of the interior angles:

n = 9

(n - 2)180° = (9 - 2)180° = (7)180° = 1260°

Then solve for the unknown angle measure, x°:

125 + 130 + 172 + 98 + 200 + 102 + 140 + 135 + x = 1260

x = 158

The unknown angle measure is 158°.

135° x°

125°

130°

172°200°140°

102°

98°

n - 2

(n - 2) 180°

Since the polygon is regular, you can divide the sum by 6 to determine each interior angle

measure.

Write and solve an equation for n, where (n - 2) 180° = 3240°.




COLLABORATIVE LEARNING

Small Group ActivityGeometry software allows students to explore the theorems in this lesson. For the Triangle Sum Theorem and the Exterior Angle Theorem, students should construct a triangle, measure the three angles, and use the Calculate tool (in the Measure menu) to find the sum of the interior angle measures and also to find the sum of the exterior angles. As students drag the vertices of the triangle to change its shape, the individual angle measures will change, but the sum of the measures will remain 180° for the interior angles and 360° for the exterior angles.

QUESTIONING STRATEGIESHow do you use the sum of the angles of a triangle to find the sum of the interior angle

measures of a convex polygon? If a convex polygon

is broken up into triangles, then the sum of the

interior angles is the number of triangles times 180°.

EXPLAIN 1 Using Interior Angles

INTEGRATE MATHEMATICAL PRACTICESFocus on Math ConnectionsMP.1 You may want to review with students how to evaluate algebraic expressions and how to use inverse operations to solve equations.

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B Determine the unknown interior angle measure of a convex octagon in which the measures of the seven other angles have a sum of 940°.

n =

Sum = ( - 2 ) 180° = ( ) 180° =

+ x =

x =

The unknown angle measure is .

Reflect

5. How might you use the Polygon Angle Sum Theorem to write a rule for determining the measure of each interior angle of any regular convex polygon with n sides?

Your Turn

6. Determine the unknown angle measures in this pentagon.

7. Determine the measure of the fourth interior angle of a quadrilateral if you know the other three measures are 89°, 80°, and 104°.

8. Determine the unknown angle measures in a hexagon whose six angles measure 69°, 108°, 135°, 204°, b°, and 2b°.

x°x°

1080°

1080

8

8

940

140

140°

6

You can divide the angle sum by n. (n - 2) 180°

__ n gives the measure of an interior angle for

any regular polygon.

n = 5

Sum = (5 - 2) 180° = (3) 180° = 540°

270 + 2x = 540

2x = 270

x = 135

Each unknown angle measure is 135°.

n = 4

Sum = (4 - 2) 180° = 2 (180°) = 360°

89 + 80 + 104 + x = 360

x = 87

The unknown angle measure is 87°.

n = 6

Sum = (6 - 2) 180° = (4) 180° = 720°

b + 2b + 69 + 108 + 135 + 204 = 720

3b + 516 = 720

3b = 204

b = 68

2b = 136

The two unknown angle measures are 68° and 136°.





DIFFERENTIATE INSTRUCTION

ManipulativesHave students fold and crease the four corners of a sheet of paper. Next, ask them to open the folds to reveal a creased polygon shape. Have students classify the polygon (octagon). Ask them to find the sum of the interior and exterior angle measures (1080°; 360°). Then have students measure the interior and exterior angles to verify their sums.

QUESTIONING STRATEGIESHow do you use the sum of the interior angle measures of a polygon to find the measure of

an unknown interior angle? Use the Polygon Sum

Theorem to find the total measure of the interior

angles, then solve an algebraic equation to find the

unknown angle.



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Explain 2 Proving the Exterior Angle TheoremAn exterior angle is an angle formed by one side of a polygon and the extension of an adjacent side. Exterior angles form linear pairs with the interior angles.

A remote interior angle is an interior angle that is not adjacent to the exterior angle.

Example 2 Follow the steps to investigate the relationship between each exterior angle of a triangle and its remote interior angles.

Step 1 Use a straightedge to draw a triangle with angles 1, 2, and 3. Line up your straightedge along the side opposite angle 2. Extend the side from the vertex at angle 3. You have just constructed an exterior angle. The exterior angle is drawn supplementary to its adjacent interior angle.

Step 2 You know the sum of the measures of the interior angles of a triangle.

m∠1 + m∠2 + m∠3 = °

Since an exterior angle is supplementary to its adjacent interior angle, you also know:

m∠3 + m∠4 = °

Make a conjecture: What can you say about the measure of the exterior angle and the measures of its remote interior angles?

Conjecture:

The conjecture you made in Step 2 can be formally stated as a theorem.

Exterior Angle Theorem

The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles.

Step 3 Complete the proof of the Exterior Angle Theorem.

∠4 is an exterior angle. It forms a linear pair with interior angle ∠3. Its remote interior angles are ∠1 and ∠2.

By the , m∠1 + m∠2 + m∠3 = 180°.

Also, m∠3 + m∠4 = because they are supplementary and make a straight angle.

By the Substitution Property of Equality, then, m∠1 + m∠2 + m∠3 = m∠ + m∠ .

Subtracting m∠3 from each side of this equation leaves .

This means that the measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.

Exterior angle

Remoteinteriorangles

1

2

3 4

1

2

3 4

180

180

The measure of the exterior angle is the same as the sum of the measures of

its two remote interior angles.

Triangle Sum Theorem

m∠1 + m∠2 = m∠4

180°

3 4




EXPLAIN 2 Proving the Exterior Angle Theorem

QUESTIONING STRATEGIESHow does finding the measure of an exterior angle differ from finding the measure of an

interior angle? The measure of an exterior angle is

the supplement of its adjacent interior angle

because the angles form linear pairs with the

interior angles. The measure of an interior angle is

not found by using linear pairs.

Why is the Exterior Angle Theorem sometimes called a corollary of the Triangle

Sum Theorem? because the Exterior Angle

Theorem follows from the Triangle Sum Theorem

LANAGUAGE SUPPORT

Communicate MathHave students write clues about interior and exterior angles in polygons, for example: “The sum of the interior angles of this three-sided polygon is 180 degrees” or “The sum of the exterior angles of this three-sided polygon is 360 degrees.” Have each student write two clue cards about different polygons. They then read their clues to the rest of the group, and the group must decide which polygon fits the clue.

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Reflect

9. Discussion Determine the measure of each exterior angle. Add them together. What can you say about their sum? Explain.

10. According to the definition of an exterior angle, one of the sides of the triangle must be extended in order to see it. How many ways can this be done for any vertex? How many exterior angles is it possible to draw for a triangle? for a hexagon?

Explain 3 Using Exterior AnglesYou can apply the Exterior Angle Theorem to solve problems with unknown angle measures by writing and solving equations.

Example 3 Determine the measure of the specified angle.

Find m∠B.

Write and solve an equation relating the exterior and remote interior angles.

145 = 2z + 5z - 2

145 = 7z - 2

z = 21

Now use this value for the unknown to evaluate the expression for the required angle.

m∠B = (5z - 2)° = (5(21) - 2)°

= (105 - 2)°

= 103°

Find m∠PRS.

Write an equation relating the exterior and remote interior angles.

Solve for the unknown.

Use the value for the unknown to evaluate the expression for the required angle.

60° 40°

145° (5z - 2)°

2z°

B

A

CD

(3x - 8)°(x + 2)°

P

Q

R

S

The exterior angles will measure 140°, 120°, and 100°.

Their sum is 360°. Each exterior angle is equal to the sum

of the measures of the two remote interior angles, and

the sum of all 3 exterior angles includes each interior

angle twice.

Two exterior angles can be drawn from any vertex by extending either side, so a triangle

can have 6 exterior angles. You could draw 12 different exterior angles for a hexagon.

3x - 8 = (x + 2) + 90

3x - 8 = x + 92

2x = 100

x = 50

m∠PRS = (3x - 8) ° = (3 (50) - 8) ° = 142°





AVOID COMMON ERRORSSome students may confuse the theorems in this lesson and incorrectly assume that the sum of the interior angles of a polygon is 360°. Remind students of the Triangle Sum Theorem. Have them draw an equilateral triangle and show that its interior angle measures add to 180° and its exterior angle measures add to 360°.

EXPLAIN 3 Using Exterior Angles

INTEGRATE MATHEMATICAL PRACTICESFocus on PatternsMP.8 Encourage students to make a table listing the sums of the exterior angles of regular triangles, quadrilaterals, pentagons, and hexagons. Ask them what they notice about the sum of the exterior angles. (The sum is always 360°.) Ask them to find the pattern in the measure of each individual exterior angle for these regular polygons. (They each have the same measure.)

CONNECT VOCABULARY Help students understand the meanings of interior, exterior, and remote by writing the definitions on note cards. An interior angle is inside the figure, an exterior angle is outside the figure, and a remote interior angle is interior and away from the exterior angle. Relate the idea of a remote interior angle to a television remote control that sends a signal across the room and away from you.



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Your Turn

Determine the measure of the specified angle.

11. Determine m∠N in △MNP. 12. If the exterior angle drawn measures 150°, and the measure of ∠D is twice that of ∠E, find the measure of the two remote interior angles.

Elaborate

13. In your own words, state the Polygon Angle Sum Theorem. How does it help you find unknown angle measures in polygons?

14. When will an exterior angle be acute? Can a triangle have more than one acute exterior angle? Describe the triangle that tests this.

15. Essential Question Check-In Summarize the rules you have discovered about the interior and exterior angles of triangles and polygons.

(3x + 7)°

63° (5x + 50)°

M P Q

N

150°

E F G

D

5x + 50 = (3x + 7) + 63

5x + 50 = 3x + 70

2x = 20

x = 10

m∠N = (3x + 7) ° = (3 (10) + 7) ° = 37°

x + 2x = 150

3x = 150

x = 50

m∠E = x° = 50°

m∠D = 2x° = 100°

Possible answer: The sum of the measures of the interior angles of a convex polygon

equals 180 (n - 2) °. You can use it to find an unknown measure of an interior angle of a

polygon when you know the measures of the other angles.

An exterior angle will be acute when paired with an obtuse adjacent interior angle;

therefore, the triangle must be obtuse. Since a triangle must have two or three acute

interior angles, at least two exterior angles must be obtuse.

The sum of the measures of the interior angles of a triangle is 180°. The sum of the

measures of the interior angles for any polygon can be found by the rule (n - 2) 180°,

where n represents the number of sides of the polygon. The measure of an exterior angle

of a triangle is equal to the sum of the measures of the two remote interior angles.




2

31

2

431

QUESTIONING STRATEGIESWhat kind of angle is formed by extending one of the sides of a triangle? What is its

relationship to the adjacent interior angle? an

exterior angle; the angles are supplementary.

ELABORATE QUESTIONING STRATEGIES

How do you use the sum of the interior angle measures of a regular polygon to find the

measure of each interior angle? Divide the sum of

the interior angles by the number of sides.

What happens to the measure of each exterior angle as the number of sides of a regular

polygon increases? Why? The measures get smaller

and smaller because the sum must remain 360°.

SUMMARIZE THE LESSONHave students fill out a chart to summarize the theorems in this lesson. Sample:

Triangle Sum Theorem m∠1 + m∠2 + m∠3 = 180°

Polygon Sum Theorem (n - 2) 180° = (6 - 2) 180° = 720°

Exterior Angle Theorem m∠4 = m∠1 + m∠2

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1. Consider the Triangle Sum Theorem in relation to a right triangle. What conjecture can you make about the two acute angles of a right triangle? Explain your reasoning.

2. Complete a flow proof for the Triangle Sum Theorem.

Given △ABC Prove m∠1 + m∠2 + m∠3 = 180°

3. Given a polygon with 13 sides, find the sum of the measures of its interior angles.

4. A polygon has an interior angle sum of 3060°. How many sides must the polygon have?

5. Two of the angles in a triangle measure 50° and 27°. Find the measure of the third angle.

Solve for the unknown angle measures of the polygon.

6. A pentagon has angle measures of 100°, 105°, 110° and 115°. Find the fifth angle measure.

7. The measures of 13 angles of a 14-gon add up to 2014°. Find the fourteenth angle measure?

• Online Homework• Hints and Help• Extra Practice

Evaluate: Homework and Practice

A C

B ℓ

1

2

3

4 5

Parallel Postulate

Draw ℓ parallel to AC through B.

m∠3 = m∠5 m∠4 + m∠2 + m∠5 = 180°

Alternate Interior Definition of straight angleAngles Theorem

Alt Int Angles Theorem

m∠1 + m∠2 + m∠3 =180°


m∠1 = m∠4

They must be complementary. One angle of the right triangle measures 90°. So the sum of the remaining two angles is 180° - 90° = 90°.

(n - 2) 180° = (13 - 2) 180° = (11) 180° = 1980°

A polygon with 13 sides has an interior angle measure sum of 1980°.

3060 = (n - 2) 180

19 = n

The polygon must have 19 sides.

50 + 27 + x = 180

x = 103

The measure of the third angle is 103°.

(5 - 2) 180° = (3) 180° = 540°

540 = 100 + 105 + 110 + 115 + x

110 = x

The measure of the fifth angle is 110°.

(14 - 2) 180° = (12) 180° = 2160°

2014 + x = 2160

x = 146

The measure of the 14th angle is 146°.




GE_MNLESE385795_U2M07L1.indd 320 20/03/14 12:28 PMExercise Depth of Knowledge (D.O.K.)COMMONCORE Mathematical Practices

1–2 2 Skills/Concepts MP.2 Reasoning

3–5 2 Skills/Concepts MP.6 Precision

6–9 2 Skills/Concepts MP.5 Using Tools

10–26 2 Skills/Concepts MP.4 Modeling

27 3 Strategic Thinking MP.3 Logic

28 3 Strategic Thinking MP.6 Precision

29 3 Strategic Thinking MP.2 Reasoning

EVALUATE

ASSIGNMENT GUIDE

Concepts and Skills Practice

Explore 1Exploring Interior Angles in Triangles

Exercises 1–2

Explore 2Exploring Interior Angles in Polygons

Exercises 3–5

Example 1Using Interior Angles

Exercises 6–9

Example 2Proving the Exterior Angle Theorem

Exercises 10

Example 3Using Exterior Angles

Exercises 11–14

INTEGRATE MATHEMATICAL PRACTICESFocus on TechnologyMP.5 Students can use a spreadsheet as a reference to find the sum of the interior angles of a convex polygon with n sides for n = 3 to 30 (or higher). They could also use the spreadsheet to give the measure of each interior and exterior angle of a regular polygon with n sides.



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8. Determine the unknown angle measures for the quadrilateral in the diagram.

9. The cross-section of a beehive reveals it is made of regular hexagons. What is the measure of each angle in the regular hexagon?

10. Create a flow proof for the Exterior Angle Theorem.

Find the value of the variable to find the unknown angle measure(s).

11. Find w to find the measure of the exterior angle. 12. Find x to find the measure of the remote interior angle.

3x°4x°

x° 2x°

1

2

3 4

m∠1 + m∠2 + m∠3 = 180°

Triangle Sum Theorem

m∠1 + m∠2 + m∠3 = m∠3 + m∠4


m∠1 + m∠2 = m∠4

Substraction Property of Equality

m∠3 + m∠4 = 180°

Definition of supplementary

w°

68°

x°

134°46°

(4 - 2) 180° = (2) 180° = 360° 2x = 72

x + 2x + 3x + 4x = 360 3x = 108

x = 36 4x = 144The measures of the interior angles of the quadrilateral are 36°, 72°, 108°, and 144°.

(n - 2) 180° = (6 - 2) 180° = (4) 180° = 720°

6x = 720

x = 120

Each angle of a regular hexagon measures 120°.

w = 68 + 68

w = 136

x + 46 = 134

x = 88




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13. Find m∠H.

14. Determine the measure of the indicated exterior angle in the diagram.

15. Match each angle with its corresponding measure, given m∠1 = 130° and m∠7 = 70°. Indicate a match by writing the letter for the angle on the line in front of the corresponding angle measure.

A. m∠2 50°

B. m∠3 60°

C. m∠4 70°

D. m∠5 110°

E. m∠6 120°

16. The map of France commonly used in the 1600s was significantly revised as a result of a triangulation survey. The diagram shows part of the survey map. Use the diagram to find the measure of ∠KMJ. Note that ∠KMJ ≅ ∠NKM.

17. An artistic quilt is being designed using computer software. The designer wants to use regular octagons in her design. What interior angle measures should she set in the computer software to create a regular octagon?

(3x + 4)° ?

3x°

2x°

123

56

7

4

70°

104°

88° 48°

126°

F G

H

J

(5x + 17)°

(6x - 1)°

(6x - 1) + (5x + 17) = 126

x = 10

m∠H = (6x - 1) ° = (6 (10) - 1) ° = 59°

180 - (3x + 4) = 2x + 3x

22 = x

180 - (3 (22) + 4) = 180 - (66 + 4) = 180 - 70 = 110

The measure of the indicated exterior angle is 110°.

A

B

D

E

C

m∠KMN + m∠MNK + m∠NKM = 180°

88° + 48° + m∠NKM = 180°

136° + m∠NKM = 180°

m∠NKM = 44°

∠KMJ ≅ ∠NKM, so m∠KMJ = m∠NKM = 44°.

(n - 2) 180° = (8 - 2) 180° = (6) 180° = 1080°

1080° _ 8

= 135°

The designer should set the interior angles of the regular octagon at 135°.


DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-D;CA-D

DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-D;CA-D


INTEGRATE MATHEMATICAL PRACTICESFocus on CommunicationMP.3 Discuss each of the following questions about triangles as a class. Have students explain how the Triangle Sum Theorem justifies their responses.

1. A triangle can have only one obtuse angle or only one right angle.

2. The acute angles of a right triangle are complementary.



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18. A ladder propped up against a house makes a 20° angle with the wall. What would be the ladder's angle measure with the ground facing away from the house?

19. Photography The aperture of a camera is made by overlapping blades that form a regular decagon.

a. What is the sum of the measures of the interior angles of the decagon?

b. What would be the measure of each interior angle? each exterior angle?

c. Find the sum of all ten exterior angles.

20. Determine the measure of ∠UXW in the diagram.

21. Determine the measures of angles x, y, and z.

20°

?

hous

e

ground

ladd

er

78°

54°

X

Y

Z

UW

V

55°

80°100°

60°x°z°y°

The house is perpendicular to the ground, so the other remote interior angle is 90°. 20 + 90 = 110, so the measure of the indicated exterior angle is 110°.

(10 - 2) 180° = (8) 180° = 1440°

1440° ÷ 10 = 144°; 180° - 144° = 36°

36° (10) = 360°

m∠WUX = 90°

180° = 54° + 90° + m∠UXW

m∠UXW = 36°

x = 180 - (100 + 60) = 20°

y = 180 - (80 + 55) = 45°

z = 180 - (20 + 45) = 115°




323 Lesson 7 . 1


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22. Given the diagram in which ‾→ BD bisects ∠ABC and

‾→ CD bisects

∠ACB, what is m∠BDC?

23. What If? Suppose you continue the congruent angle construction shown here. What polygon will you construct? Explain.

24. Algebra Draw a triangle ABC and label the measures of its angles a°, b°, and c°. Draw ray BD that bisects the exterior angle at vertex B. Write an expression for the measure of angle CBD.

25. Look for a Pattern Find patterns within this table of data and extend the patterns to complete the remainder of the table. What conjecture can you make about polygon exterior angles from Column 5?

Column 1 Number of

Sides

Column 2 Sum of the

Measures of the Interior Angles

Column 3 Average

Measure of an Interior Angle

Column 4 Average

Measure of an Exterior Angle

Column 5 Sum of the Measures of the Exterior

Angles

3 180° 60° 120° 120° (3) =

4 360° 90° 90° 90° (4) =

5 540° 108°

6 120°

Conjecture:

15°

B

A C

D

120°a°

c°

b°A

C

D

B

m∠ABC = 2 (m∠DBC) = 2 (15°) = 30°

30° + m∠ACB + 90° = 180°, so m∠ACB = 60°.

Then, m∠DCB = 1 _ 2

(m∠ACB) = 1 _ 2

(60°) = 30°.

15° + m∠BDC + 30° = 180°, so m∠BDC = 135°.

A regular hexagon; if the construction continues and the sides are kept congruent, the polygon will include six 120° angles and six congruent sides, so it is a regular hexagon.

Possible answer:

m∠CBD = ( a + c ____ 2 ) °

720°

72° (5) = 360°

60° (6) = 360°

360°

360°

60°

72°

It appears from the table that the sum of the measures of the exterior angles of any polygon is always 360°.


DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-B;CA-B

DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-B;CA-B

GE_MNLESE385795_U2M07L1 324 5/22/14 5:41 PM

AVOID COMMON ERRORSSome students may multiply the number of sides of a polygon by 180 to find the sum of the interior angles of the polygon. Remind them that the sum is based on the number of triangles. Since the sum of the angles of a triangle (3 sides) is 180°, to find the sum of the interior angles, they must subtract 2 from the number of sides before multiplying by 180.



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26. Explain the Error Find and explain what this student did incorrectly when solving the following problem.

What type of polygon would have an interior angle sum of 1260°?

1260 = (n - 2) 180 7 = n - 2 5 = n

The polygon is a pentagon.

H.O.T. Focus on Higher Order Thinking

27. Communicate Mathematical Ideas Explain why if two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are also congruent.

Given: ∠L ≅ ∠R, ∠M ≅ ∠S

Prove: ∠N ≅ ∠T

28. Analyze Relationships Consider a right triangle. How would you describe the measures of its exterior angles? Explain.

29. Look for a Pattern In investigating different polygons, diagonals were drawn from a vertex to break the polygon into triangles. Recall that the number of triangles is always two less than the number of sides. But diagonals can be drawn from all vertices. Make a table where you compare the number of sides of a polygon with how many diagonals can be drawn (from all the vertices). Can you find a pattern in this table?

T S

MN

LR

The error is that the student subtracted 2 from both sides instead of adding 2. The value of n should be 9, and the polygon is a nonagon.

By the Triangle Sum Theorem, m∠L + m∠M + m∠N = 180° and m∠R + m∠S + m∠T = 180°. Since each set of angle measures total 180°, they are equal using the substitution property of equality. So, m∠L + m∠M + m∠N = m∠R + m∠S + m∠T. Since ∠L ≅ ∠R and ∠M ≅ ∠S, then m∠L = m∠R and m∠M = m∠S by the definition of congruence. Subtracting equals from both sides gives m∠N = m∠T. Then ∠N ≅ ∠T by the definition of congruence.

An exterior angle will be right when paired with a right adjacent interior angle. There can be only one right angle in a triangle. Since a triangle must have two or three acute interior angles, the other two exterior angles must be obtuse.

Number of Sides, n 3 4 5 6 7 8

Number of Diagonals, d 0 2 5 9 14 20

The number of diagonals increases by 2, then 3, 4, 5, etc. A formula

relating n and d is d = n (n - 3)

_ 2

.




JOURNALHave students review the Polygon Angle Sum Theorem and the Polygon Exterior Angle Theorem, and then draw a pentagon and show how to find its interior angle sum measures and its exterior angle sum measures. 540°, 360°

325 Lesson 7 . 1


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Lesson Performance TaskYou’ve been asked to design the board for a new game called Pentagons. The board consists of a repeating pattern of regular pentagons, a portion of which is shown in the illustration. When you write the specifications for the company that will make the board, you include the measurements of ∠BAD, ∠ABC, ∠BCD and ∠ADC. Find the measures of those angles and explain how you found them.

A

C

DB

108°

m∠BAD = m∠BCD = 36°

m∠ABC = m∠ADC = 144°

To find the measure of each interior angle of one of the pentagons, divide it

into three triangles. This gives the sum of the measures of the five angles of

the pentagon, 3 × 180° = 540°. Each angle measures 540° ÷ 5 = 108°.

Draw ̄ BD .

m∠ABD = 180° - 108° = 72°.

m∠ABC = m∠ADC = 2 × 72° = 144°

m∠BAD = m∠BCD = 180° - (72° + 72°) = 180° - 144° = 36°




EXTENSION ACTIVITY

Have students design and draw game boards consisting of congruent quadrilaterals, congruent pentagons, and/or congruent hexagons. Each design should show at least two different classes of polygons (for example, quadrilaterals and hexagons) and a total of at least six polygons. Students should write the measure of each angle directly on the figures, and write elsewhere an explanation of how, without protractors, they found each measure.

INTEGRATE MATHEMATICAL PRACTICESFocus on CommunicationMP.3 Direct students’ attention to quadrilateral ABCD. Ask:

• Without knowing anything about the angles of ABCD, how could you identify the type of quadrilateral that it is? What type is it? The sides of ABCD are sides of congruent

regular pentagons, so they are congruent.

A quadrilateral with four congruent sides is

a rhombus.

• What does the type of quadrilateral that ABCD is tell you about the angles of the figure? The

opposite angles are congruent. The adjacent

angles are supplementary.

INTEGRATE MATHEMATICAL PRACTICESFocus on Critical ThinkingMP.3 The perimeter of each pentagon in the diagram is 16 cm. What is the perimeter of quadrilateral ABCD? Explain. 12.8 cm; length of

each side of each pentagon = 16 cm ÷ 5 = 3.2 cm;

perimeter of ABCD = 4 x 3.2 = 12.8 cm

Scoring Rubric2 points: Student correctly solves the problem and explains his/her reasoning.1 point: Student shows good understanding of the problem but does not fully solve or explain.0 points: Student does not demonstrate understanding of the problem.



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