angles, triangles, and equations module 15 · 2019-02-04 · example 7k = 35 7k 7 = 35 7 k = 5...

ESSENTIAL QUESTION?

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How can you use angles, triangles, and equations to solve real-world problems?

15Angles, Triangles, and Equations

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You can find examples of triangles all around you. Some buildings, such as the Transamerica Tower, have triangular faces.

LESSON 15.1

Determining When Three Lengths Form a Triangle

6.8.A

LESSON 15.2

Sum of Angle Measures in a Triangle

6.8.A

LESSON 15.3

Relationships Between Sides and Angles in a Triangle

6.8.A

MODULE

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Complete these exercises to review skills you will need

for this chapter.

Inverse OperationsEXAMPLE 7k = 35

7k __ 7

= 35 __

7

k = 5

k + 7 = 9

k + 7 - 7 = 9 - 7

k = 2

Solve each equation using the inverse operation.

1. 9p = 54 2. m - 15 = 9

3. b __ 8

= 4 4. z + 17 = 23

Name AnglesEXAMPLE

Give two names for the angle formed by the dashed rays.

5. 6. 7.

Use three points of an angle, including the vertex, to name the angle. If there is only one angle at the vertex, you can name the angle by the vertex.

Write the vertex between the other two points. ∠AMG, ∠GMA, or ∠M.

k is multiplied by 7. To solve the equation, use the inverse operation, division.

7 is added to k.

To solve the equation, use the inverse operation, subtraction.

Unit 5420

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Reading Start-Up

Active ReadingPyramid Before beginning the module,

create a pyramid to help you organize what you learn. Label each side with one of the

lesson titles from this module. As you study

each lesson, write important ideas like

vocabulary, properties, and formulas on the

appropriate side.

VocabularyReview Words✔ acute angle (ángulo

agudo) angle (ángulo) equilateral triangle

(triángulo equilátero) inequalities (desigualdad) line segments (segmentos

de línea)✔ obtuse angle (ángulo

obtuso)✔ right angle (ángulo recto) right triangle (triángulo

rectángulo) vertex (vértice)

Visualize VocabularyUse the ✔ words to complete the graphic. You will put one

word in each oval.

Understand VocabularyComplete the sentences using the review words.

1. A triangle that contains a right angle is a .

2. An has three congruent sides and three

congruent angles.

3. The sides of triangles are . Where two lines

meet to form an angle of a triangle is called a .

Types of Angles

Description Angle

angle measure > 0° and < 90°

angle measure > 90° and < 180°

angle measure = 90°

421Module 15

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47° 81°

Unpacking the TEKSUnderstanding the TEKS and the vocabulary terms in the TEKS

will help you know exactly what you are expected to learn in this

module.

What It Means to YouYou will learn to determine if three

lengths can form a triangle.

UNPACKING EXAMPLE 6.8.A

A map of a new dog park shows that it is

triangular and that the sides measure 18

yd, 37 yd, and 17 yd. Are the dimensions

possible? Explain your reasoning.

Find the sum of the lengths of each pair of sides.

Compare the sum to the third side.

18 + 37 ? > 17 18 + 17

? > 37 37 + 17

? > 18

55 > 17 ✔ 35 ≯ 34 ✘ 54 > 18 ✔

The sum of two of the given lengths is not greater than the third

length. So, the dog park cannot have these side lengths.

What It Means to YouYou will learn how to find the measure of an angle of a triangle if

you know the measures of the other two angles.

The measures of two of the angles of a triangle are 47° and 81°.

What is the measure of the third angle of the triangle?

m ∠A + m∠B + m∠C = 180°

47° + 81° + x = 180°

128° + x = 180°

x = 52°

The third angle of the triangle measures 52°.

MODULE 15

my.hrw.com

6.8.A

Extend previous knowledge of

triangles and their properties

to include the sum of angles

of a triangle, the relationship

between the lengths of sides

and measures of angles in a

triangle, and determining

when three lengths form a

triangle.

6.8.A

Extend previous knowledge of

triangles and their properties

to include the sum of angles

of a triangle, the relationship

between the lengths of sides

and measures of angles in a

triangle, and determining when

three lengths form a triangle.

Visit my.hrw.com to see all

the

unpacked.

Unit 5422

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a = 2

c = 4F

E

A C

D

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b = 3

a = 2

c = 4

F

ESSENTIAL QUESTIONHow can you use the relationship between side lengths to determine when three lengths form a triangle?

L E S S O N

15.1Determining When Three Lengths Form a Triangle

Drawing Three SidesUse geometry software to draw a triangle whose sides

have the following lengths: 2 units, 3 units, and 4 units.

Draw three line segments of 2, 3, and 4 units of length.

Let ___

AB be the base of the triangle. Place endpoint

C on top of endpoint B and endpoint E on top of

endpoint A. These will become two of the vertices

of the triangle.

Using the endpoints C and E as fixed vertices, rotate

endpoints F and D to see if they will meet in a single point.

The line segments of 2, 3, and 4 units do / do not

form a triangle.

Repeat Steps 2 and 3, but start with a different base

length. Do the line segments make the exact same

triangle as the original?

The line segments do / do not make the same triangle as the

original.

Draw three line segments of 2, 3, and 6 units. Can you form

a triangle with the given segments?

The line segments of 2, 3, and 6 units do / do not form a triangle.

A

B

C

D

E

EXPLORE ACTIVITY 6.8.A

Expressions, equations, and relationships—6.8.A Extend previous knowledge of triangles and their properties to include . . . determining when three lengths form a triangle.

423Lesson 15.1

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Reflect1. Conjecture Try to make triangles using real world objects such as

three straws of different lengths. Find three side lengths that form a

triangle and three side lengths that do not form a triangle. What do

you notice about the lengths that do not form a triangle?

EXPLORE ACTIVITY (cont’d)

Tell whether a triangle can have sides with the given lengths.

11 cm, 6 cm, 13 cm


11 + 6 ? >

13 6 + 13

? >

11 11 + 13

? >

6


17 > 13 ✓ 19 > 11 ✓ 24 > 6 ✓

The sum of any two of the given lengths is

greater than the third length.

So, a triangle can have these side lengths.

EXAMPLE 1

A

STEP 1

STEP 2

Using Triangle Side Length RelationshipsYou saw in the Explore Activity that you cannot always form a triangle from

three given line segments.

You can use this relationship to determine if given side lengths can form a triangle.

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6.8.A

4 5

7Can form a

triangle

42

7Cannot form a

triangle

Triangle Inequality

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

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5 ft, 15 ft, 9 ft


5 + 15 ? >

9 15 + 9

? >

5 5 + 9

? >

15


20 > 9 ✓ 24 > 5 ✓ 14 ≯ 15

The sum of any two of the given lengths is not greater

than the third length.

So, a triangle cannot have these side lengths.

B

STEP 1

STEP 2

2. 3 cm, 6 cm, 9 cm 3. 4 m, 5 m, 8 m

Tell whether a triangle can have sides with the given lengths. Explain.

YOUR TURN

Using Inequalities to Represent the Relationship Between Triangle Side LengthsYou can use what you know about the relationship among the lengths of the

sides of a triangle to write an inequality. Then you can use the inequality to

determine if a given value can be the length of an unknown side.

EXAMPLEXAMPLE 2

Which value could be the length of x?

x = 15 x = 10

4 + 9 > x 4 + 9 > x

4 + 9 ? >

15 4 + 9

? >

10

13 ≯ 15 13 > 10 ✓

The value that could be the length of x is x = 10.

STEP 1

STEP 2

STEP 3

Math TalkMathematical Processes


6.8.A

Explain why a triangle with sides measuring 5 in.,

5 in., and 1 foot cannot be constructed.

Explain how you know that the Triangle Inequality relationship is true for every

equilateral triangle.

Substitute each value for x.

Compare the sum to the given value of x.

Write an inequality.

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Guided Practice

Determine whether a triangle can have sides with the given lengths.

Explain. (Explore Activity and Example 1)

1. 3 cm, 10 cm, 8 cm

2. 10 ft, 10 ft, 18 ft

3. 30 in., 20 in., 40 in.

4. 16 cm, 12 cm, 3 cm

5. Which value could be the length of x?

(Example 2)

x = 29 x = 45

6. Explain how you can determine whether three metal rods can be

joined to form a triangle.

ESSENTIAL QUESTION CHECK-IN??

4. Which value could be the length of x?

x = 35 x = 13

YOUR TURN

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189.34 mi

146.43 miAustin

Houston

San Antonio

Name Class Date

7. A map of a new dog park shows that it is

triangular and that the sides measure

18.5 m, 36.9 m, and 16.9 m. Are the

dimensions correct? Explain your reasoning.

8. Choose a real world object that you can

cut into three different lengths to form a

triangle. Find three side lengths that form a

triangle and three lengths that do not form

a triangle. For each triangle, give the side

lengths and explain why those lengths do

or do not form a triangle.

Triangle 1:

Triangle 2:

9. Could the three sides of a triangular

shopping mall measure 1 _ 2 mi, 1 _

3 mi, and

1 _ 4 mi? Show how you found your answer.

10. Geography The map shows the distance

in air miles from Houston to both Austin

and San Antonio.

a. What is the greatest possible distance

from Austin to San Antonio?

b. How did you find the answer?

c. What is the least possible distance

from Austin to San Antonio?

d. How did you find the answer?

Independent Practice15.16.8.A

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11. Critical Thinking Two sides of an isosceles triangle measure 3 inches

and 13 inches respectively. Find the length of the third side. Explain

your reasoning.

12. Critique Reasoning While on a car trip with her family, Erin saw a sign that

read, “Amarillo 100 miles, Lubbock 80 miles.” She concluded that the distance

from Amarillo to Lubbock is 100 - 80 = 20 miles. Was she right? Explain.

13. Make a Conjecture Is there a value of n for which there could be a

triangle with sides of length n, 2n, and 3n? Explain.

14. Persevere in Problem Solving A metalworker cut an 8-foot length of

pipe into three pieces and welded them to form a triangle. Each of the

3 sections measured a whole number of feet in length. How long was

each section? Explain your reasoning.

FOCUS ON HIGHER ORDER THINKING

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EXPLORE ACTIVITY

ESSENTIAL QUESTION

Exploring Angles in a TriangleRecall that a triangle is a closed figure with three line

segments and three angles. The measures of the angles of a

triangle have a special relationship with one another.

Use a straightedge to draw a large triangle. Label the

angles 1, 2, and 3.

Use scissors to cut out the triangle.

Tear off the three angles. Arrange them

around a point on a line as shown.

What is the measure of the straight

angle formed by the three angles?

What is the sum of the measures of the three angles? Explain.

Compare your results with those of your classmates. What guess can

you make?

Reflect1. Justify Reasoning How can you show that your guess is correct?

A

B

C

D

E

F

How do you use the sum of angles in a triangle to find an unknown angle measure?

L E S S O N

15.2Sum of Angle Measures in a Triangle

6.8.A

Expressions, equations, and relationships—6.8.A Extend previous knowledge of triangles and their properties to include the sum of angles in a triangle …

429Lesson 15.2

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Finding an Angle Measure in a Triangle

Find the unknown angle measures.

2. 3.

x = x =

YOUR TURN

Fountain Place, shown to the right, is a 720-foot Dallas skyscraper. Find the

measure of the unknown angle in the triangle at the top of the building.

m∠1 + m∠2 + m∠3 = 180°

65° + 65° + x = 180°

130° + x = 180°

−130° −130°

x = 50°

The angle at the top of the triangle measures 50°.

EXAMPLE 1


6.8.A

Can a triangle have two obtuse angles? Why

or why not?

2

m∠1 + m∠2 + m∠3 = 180°

1 3

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

Write an equation.

Add.

Subtract 130° from both sides.

Sum of Angle Measures of a Triangle

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

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Finding Angles in an Equilateral TriangleRecall that an equilateral triangle has three congruent sides and three

congruent angles.

Find the angle measures in the equilateral triangle.

3x = 180°

3x ___ 3

= 180° ____ 3

x = 60°

Each angle in an equilateral triangle measures 60°.

Reflect4. Multiple Representations Write a different equation to find the angle

measures in Example 2. Will the answer be the same? Explain.

5. Draw Conclusions Triangle ABC is a right triangle. What conclusions

can you draw about the measures of the angles of the triangle?

EXAMPLEXAMPLE 2

Write an equation to find the unknown angle measure in each triangle.

6. The measures of two of the angles are 25° and 65°.

7. The measures of two of the angles are 60°.


YOUR TURN

6.8.A

Write an equation.

Divide both sides by 3.

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59°HF

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x 28°

33°

Guided Practice

1. The sum of the angle measures in a triangle is .

(Explore Activity)

Find the unknown angle measure in each triangle. (Examples 1 and 2)

2. m∠R + m∠S + m∠T =

+ + x =

+ x =

- -

x =

3. 4.

x = x =

5. 6.

x = x =


8. The measures of two of the angles are 50° and 30°.

9. Arlen knows the measures of two angles of a triangle. Explain how he

can find the measure of the third angle. Why does your method work?


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Name Class Date

Figure ABCD represents a garden crossed by

straight walkway ___

AC . Use the figure for 10–15.

10. Find m∠DAC.

11. Explain how you found m∠DAC.

12. Find m∠BAC.

13. Explain how you found m∠BAC.

14. Find m∠DAB.

15. Explain how you found m∠DAB.

16. An observer at point O sees airplane

P directly over airport A. The observer

measures the angle of the plane at 40.5°.

Find m∠P.

The map shows the intersection of three

streets in San Antonio’s River Walk district.

Use the map for 17–18.

17. Find the measures of the three angles of

the triangle.

18. Explain how you found the angle

measures.

15.2 Independent Practice6.8.A

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Work Area

19. Persevere in Problem Solving Find the measure of ∠ACB. Explain how

you found your answer.

20. Communicate Mathematical Ideas Explain how you can use the figure

to find the sum of the measures of the angles of quadrilateral ABCD.

What is the sum?

21. Draw Conclusions Recall that a right triangle is a triangle with one right

angle. One angle of a triangle measures 89.99 degrees. Can the triangle

be a right triangle? Explain your reasoning.


Unit 5434

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ESSENTIAL QUESTIONHow can you use the relationships between side lengths and angle measures in a triangle to solve problems?

L E S S O N

15.3Relationships Between Sides and Angles in a Triangle

Exploring the Relationship Between Sides and Angles in a Triangle There is a special relationship between the lengths of sides

and the measures of angles in a triangle.

Use geometry software to make triangle ABC.

Make ∠A the smallest angle.

Choose one vertex and drag it so that you

lengthen the side of the triangle opposite angle

A. Describe what happens to ∠A.

Drag the vertex to shorten the side opposite ∠B.

What happens to ∠B?

Make several new triangles. In each case, note

the locations of the longest and shortest sides

in relation to the largest and smallest angles.

Describe your results.

A

B

C

D

EXPLORE ACTIVITY 6.8.A

Expressions, equations, and relationships—6.8.A Extend previous knowledge of triangles and their properties to include…the relationship between the lengths of sides and measures of angles in a triangle…

435Lesson 15.3

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Using the Relationship Between Sides and Angles in a Triangle You have seen that in a triangle the largest angle is opposite the longest side

and the smallest angle is opposite the shortest side. It follows that the midsize

angle is opposite the midsize side.

Triangle ABC has side lengths of 7 cm, 9 cm,

and 4.5 cm. Use the relationship between the

sides and angles of a triangle to match each

side with its correct length.

AC = 9 cm

AB = 4.5 cm

BC = 7 cm

Triangle ABC has angles measuring 60°, 80°, and

40°. Use the relationship between the sides and

angles of a triangle to match each angle with its

correct measure.

m∠A = 80°

m∠C = 40°

m∠B = 60°

EXAMPLE 1

A

B

1. Triangle ABC has side lengths of 11, 16, and 19.

Match each side with its correct length.

AB = AC = BC =

2. Triangle ABC has angle measures of 45°, 58°, and 77°.

Match each angle with its correct measure.

m∠A = m∠B = m∠C =

YOUR TURN

6.8.A

The longest side is opposite the largest angle.

The shortest side is opposite the smallest angle.

The midsize side is opposite the midsize angle.

The largest angle is opposite the longest side.The smallest angle is opposite the shortest side.

The midsize angle is opposite the midsize side.

Unit 5436

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FPO

90°

45°

45°

longest side

90°

45°

45°

Solving Problems Using Triangle Relationships Recall that triangles can be classified by the lengths of their sides. A scalene

triangle has no congruent sides. An isosceles triangle has two congruent sides.

An equilateral triangle has three congruent sides.

Brandy is making a quilt. Each block of the quilt is made up of four

triangles. Each triangle is in the shape of a right isosceles triangle.

Two of the side measures of one triangle are 6.4 inches and 9 inches.

Brandy wants to add a ribbon border around one of the triangles. How

much ribbon will she need?

Analyze Information

Rewrite the question as a statement.

• Find the amount of ribbon Brandy will need for a border around one

triangle.

Identify the important information.

• Each quilt piece has the shape of a right isosceles triangle.

• Two sides of the triangle measure 6.4 inches and 9 inches.

Formulate a Plan

You can draw a model and label it with the important information to find

the total length of ribbon that Brandy needs for one triangle.

Justify and EvaluateSolve

Think: A right triangle will have one 90° angle.

Since the sum of the angles is 180°, the other

two angles will be congruent and will have a

combined measure of 90°.

90° ÷ 2 = 45°

Label the new information on the model.

90° is the greatest angle measure, so the side

opposite the 90° angle will be the longest side.

The other two angles are congruent, so the sides

opposite those angles are congruent.

The shortest side lengths are 6.4 inches and

6.4 inches. So, Brandy will need

6.4 + 6.4 + 9 = 21.8 inches of ribbon.

Justify and Evaluate

The solution is reasonable because the quilt piece is in the shape of an

isosceles right triangle and it has two sides measuring 6.4 inches and 9 inches.

EXAMPLEXAMPLE 2 ProblemSolving 6.8.A

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105°

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20 in.

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1. Triangle ABC has side lengths of 17, 13, and 24. Match

each side with its correct length. (Example 1)

= 24 = 13 = 17

2. The figure represents a traffic island that has angles

measuring 60°, 20°, and 100°. Match each angle with

its correct measure. (Example 1)

m∠ = 100° m∠ = 20° m∠ = 60°

3. Vocabulary Explain how the relationship between

the sides and angles of a triangle applies to equilateral

triangles. (Example 2)

4. Ramone is building a fence around a vegetable garden in his backyard.

The fence will be in the shape of a right isosceles triangle. Two of the side

measures are 12 feet and 16 feet. Use a problem solving model to find

the total length of fencing he needs. Explain. (Example 2)

5. Describe the relationship between the lengths of the sides and the

measures of the angles in a triangle.


Guided Practice

3. A fence around a rock garden is in the shape of a right triangle. Two

angles measure 30° and 60°. Two sides measure 10 feet and 17.3 feet.

The total length of the fence is 47.3 feet. How long is the side opposite

the right angle?

YOUR TURN

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Name Class Date

Independent Practice15.3

Use the figure for 6–8.

6. Critique Reasoning Dustin says that

△FGH is an equilateral triangle because

the sides appear to be the same length. Is

his reasoning valid? Explain.

7. What additional information do you need

to know before you can determine which

side of the triangle is the longest? How can

you find it?

8. Which side of the triangle is the longest?

Explain how you found the answer.

The figure shows the angle measurements

formed by two fenced-in animal pens that

share a side. Use the figure for 9–10.

9. Caitlin says that _ AC is the longest segment

of fencing because it is opposite 68°, the

largest angle measure in the figure. Is her

reasoning valid? Explain.

10. What is the longest segment of fencing in

△ABC? Explain your reasoning.

11. Find the longest segment of fencing in the

figure. Explain your reasoning.

6.8.A

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30°30°

12. In triangle ABC, _ AB is longer than

_ BC and

_ BC is longer than

_ AC .

a. Draw a sketch of triangle ABC.

b. Name the smallest angle in the triangle. Explain your reasoning.

13. Persevere in Problem Solving

Determine the shortest line segment in the

figure. Explain how you found the answer.

14. Communicate Mathematical Ideas Explain how the relationship

between the sides and angles of a triangle applies to isosceles triangles.

15. Critical Thinking Can a scalene triangle contain a pair of congruent

angles? Explain.


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112°

41°88°

38°

35° 56°

89°

A

BC

12 7

16

F

ED

MODULE QUIZ

15.1 Determining When Three Lengths Form a TriangleDetermine whether the three side lengths form a triangle.

1. 3, 5, 7 2. 9, 15, 4

3. 17, 5, 23 4. 28, 16, 38

15.2 Sum of Angle Measures in a TriangleFind the unknown angle measures.

5. 6.

15.3 Relationships Between Sides and Angles in a TriangleMatch each of the given measures to the correct side or angle.

7. 11, 7.5, 13

8. 24°, 44°, 112°

9. How can you describe the relationships among angles and sides in

a triangle?

ESSENTIAL QUESTION

441Module 15

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X

Z Y29°

133°

A

C B43° 87°

50°

29

22

4x

F

D E65° 56°

Selected Response

1. The two longer sides of a triangle measure 16 and 22. Which of the following is a possible length of the shortest side?

A 4 C 11

B 6 D 19

2. Part of a large metal sculpture will be a triangle formed by welding three bars together. The artist has four bars that measure 12 feet, 7 feet, 5 feet, and 3 feet. Which bar could not be used with two of the others to form a triangle?

A the 3-foot bar

B the 5-foot bar

C the 7-foot bar

D the 12-foot bar

3. What is the measure of the missing angle in the triangle below?

A 39° C 59° B 49° D 69°

4. The measure of ∠A in △ABC is 88°. The measure of ∠B is 60% of the measure of ∠A. What is the measure of ∠C?

A 39.2° C 91° B 52.8° D 127.2°

5. Which of these could be the value of x in the triangle below?

A 5

B 6

C 7

D 10

Gridded Response

6. Find m∠Z.

. + 0 0 0 0 0 0

- 1 1 1 1 1 1

2 2 2 2 2 2

3 3 3 3 3 3

4 4 4 4 4 4

5 5 5 5 5 5

6 6 6 6 6 6

7 7 7 7 7 7

8 8 8 8 8 8

9 9 9 9 9 9

Module 15 MIXed ReVIeW

Texas Test Prep

442 Unit 5

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DO NOT EDIT--Changes must be made through “File info” CorrectionKey=C

angles, triangles, and equations module 15 · 2019-02-04 · example 7k = 35 __ 7k 7 = 35__ 7 k = 5...

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angles, triangles, and equations module 15 · 2019-02-04 · example 7k = 35 7k 7 = 35 7 k = 5...