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Angles, Triangles,
and Equations
?
MODULE
15
LESSON 15.1
ESSENTIAL QUESTION
Determining When
Three Lengths Form
a Triangle
How can you use angles,
triangles, and equations
to solve real-world
problems?
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
© Houghton Mifflin Harcourt Publishing Company • Image Credits: Richard Nowitz/
Photodisc/Getty Images
6.8.A
Real-World Video
You can find examples of triangles all around you.
Some buildings, such as the Transamerica Tower,
have triangular faces.
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Math On the Spot
Animated Math
Personal Math Trainer
Go digital with your
write-in student
edition, accessible on
any device.
Scan with your smart
phone to jump directly
to the online edition,
video tutor, and more.
Interactively explore
key concepts to see
how math works.
Get immediate
feedback and help as
you work through
practice sets.
419
Are YOU Ready?
Personal
Math Trainer
Complete these exercises to review skills you will need
for this chapter.
Inverse Operations
EXAMPLE
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7k = 35
Online
Assessment and
Intervention
k is multiplied by 7.
To solve the equation, use the
inverse operation, division.
35
7k
__
= __
7
7
k=5
7 is added to k.
k+7=9
k+7-7=9-7
k= 2
To solve the equation, use the inverse
operation, subtraction.
Solve each equation using the inverse operation.
1. 9p = 54
2. m - 15 = 9
3. __b8 = 4
4. z + 17 = 23
Name Angles
EXAMPLE
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.
A
M
G
5.
6.
Z
7.
N
B
S
K
420
Unit 5
T
J
V
M
R
P
B
T
F
L
© Houghton Mifflin Harcourt Publishing Company
Give two names for the angle formed by the dashed rays.
Reading Start-Up
Vocabulary
Review 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 Vocabulary
Use the ✔ words to complete the graphic. You will put one
word in each oval.
Types of Angles
Description
Angle
angle measure
> 0° and < 90°
angle measure
> 90° and < 180°
angle measure = 90°
Understand Vocabulary
Complete the sentences using the review words.
1. A triangle that contains a right angle is a
2. An
congruent angles.
has three congruent sides and three
3. The sides of triangles are
© Houghton Mifflin Harcourt Publishing Company
.
meet to form an angle of a triangle is called a
. Where two lines
.
Active Reading
Pyramid 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.
Module 15
421
MODULE 15
Unpacking the TEKS
Understanding the TEKS and the vocabulary terms in the TEKS
will help you know exactly what you are expected to learn in this
module.
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.
What It Means to You
You 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 ✔
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.
What It Means to You
You 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?
x
47° + 81° +
128°
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the
unpacked.
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422
Unit 5
C
m∠A + m∠B + m∠C = 180°
+
x
= 180°
x = 180°
x = 52°
47°
A
The third angle of the triangle measures 52°.
81°
B
© Houghton Mifflin Harcourt Publishing Company • Image Credits: Thinkstock/
Comstock Images/Getty Images
The sum of two of the given lengths is not greater than the third
length. So, the dog park cannot have these side lengths.
LESSON
15.1
?
Determining When
Three Lengths Form
a Triangle
ESSENTIAL QUESTION
Expressions,
equations, and
relationships—
6.8.A Extend previous
knowledge of triangles
and their properties to
include . . . determining when
three lengths form a triangle.
How can you use the relationship between side lengths to
determine when three lengths form a triangle?
EXPLORE ACTIVITY
6.8.A
Drawing Three Sides
Use geometry software to draw a triangle whose sides
have the following lengths: 2 units, 3 units, and 4 units.
E
F
c=4
A Draw three line segments of 2, 3, and 4 units of length.
C
D
b=3
A
B
a=2
___
B 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.
F
D
c=4
E
B
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A a=2 C
C 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.
D Repeat Steps 2 and 3, but start with a different base
length. Do the line segments make the exact same
triangle as the original?
D
c=4
E
A a=2 C
b=3
F
b=3
B
The line segments do / do not make the same triangle as the
original.
E 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.
Lesson 15.1
423
EXPLORE ACTIVITY (cont’d)
Reflect
1.
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?
Using Triangle Side Length
Relationships
Math On the Spot
You saw in the Explore Activity that you cannot always form a triangle from
three given line segments.
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Triangle Inequality
The sum of the lengths of
any two sides of a triangle
is greater than the length
of the third side.
4
5
7
Can form a
triangle
4
2
7
Cannot form a
triangle
EXAMPLE 1
6.8.A
Tell whether a triangle can have sides with the given lengths.
Animated
Math
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A 11 cm, 6 cm, 13 cm
STEP 1
Find the sum of the lengths of each pair of sides.
?
?
?
11 + 6 > 13 6 + 13 > 11 11 + 13 > 6
STEP 2
Compare the sum to the third side.
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.
424
Unit 5
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You can use this relationship to determine if given side lengths can form a triangle.
B 5 ft, 15 ft, 9 ft
STEP 1
Find the sum of the lengths of each pair of sides.
?
?
?
15 + 9 > 5 5 + 9 > 15
5 + 15 > 9
STEP 2
Compare the sum to the third side.
20 > 9 ✓
24 > 5 ✓
Math Talk
14 ≯ 15
Mathematical Processes
Explain why a triangle
with sides measuring 5 in.,
5 in., and 1 foot cannot
be constructed.
The sum of any two of the given lengths is not greater
than the third length.
So, a triangle cannot have these side lengths.
YOUR TURN
Tell whether a triangle can have sides with the given lengths. Explain.
2. 3 cm, 6 cm, 9 cm
3.
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4 m, 5 m, 8 m
Online Assessment
and Intervention
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Using Inequalities to Represent
the Relationship Between Triangle
Side Lengths
Math On the Spot
You 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.
EXAMPL 2
EXAMPLE
6.8.A
Which value could be the length of x?
x = 15
x = 10
4
9
STEP 1
4+9>x
4+9>x
x
Write an inequality.
STEP 2
?
4 + 9 > 15
?
4 + 9 > 10
Substitute each value for x.
STEP 3
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Math Talk
Mathematical Processes
Explain how you know
that the Triangle Inequality
relationship is true for every
equilateral triangle.
Compare the sum to the
given value of x.
The value that could be the length of x is x = 10.
13 ≯ 15
13 > 10 ✓
Lesson 15.1
425
YOUR TURN
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Math Trainer
4. Which value could be the length of x?
Online Assessment
and Intervention
20
x = 35
13
x = 13
x
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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
x = 29
x = 45
17
x
22
?
?
ESSENTIAL QUESTION CHECK-IN
6. Explain how you can determine whether three metal rods can be
joined to form a triangle.
426
Unit 5
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5. Which value could be the length of x?
(Example 2)
Name
Class
Date
15.1 Independent Practice
Personal
Math Trainer
6.8.A
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.
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Online
Assessment and
Intervention
10. Geography The map shows the distance
in air miles from Houston to both Austin
and San Antonio.
Austin
146.43 mi
Houston
San Antonio
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.
189.34 mi
a. What is the greatest possible distance
from Austin to San Antonio?
b. How did you find the answer?
Triangle 1:
© Houghton Mifflin Harcourt Publishing Company
c. What is the least possible distance
from Austin to San Antonio?
Triangle 2:
d. How did you find the answer?
9. Could the three sides of a triangular
shopping mall measure _12 mi, _13 mi, and
_1 mi? Show how you found your answer.
4
Lesson 15.1
427
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.
FOCUS ON HIGHER ORDER THINKING
Work Area
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.
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.
428
Unit 5
© Houghton Mifflin Harcourt Publishing Company
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.
LESSON
15.2
?
Sum of Angle
Measures in a
Triangle
ESSENTIAL QUESTION
Expressions,
equations, and
relationships—6.8.A
Extend previous knowledge of
triangles and their properties
to include the sum of angles
in a triangle …
How do you use the sum of angles in a triangle to find an
unknown angle measure?
EXPLORE ACTIVITY
6.8.A
Exploring Angles in a Triangle
Recall 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.
2
A Use a straightedge to draw a large triangle. Label the
angles 1, 2, and 3.
3
1
B Use scissors to cut out the triangle.
C Tear off the three angles. Arrange them
around a point on a line as shown.
1
2
3
D What is the measure of the straight
angle formed by the three angles?
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E What is the sum of the measures of the three angles? Explain.
F Compare your results with those of your classmates. What guess can
you make?
Reflect
1. Justify Reasoning How can you show that your guess is correct?
Lesson 15.2
429
Finding an Angle Measure in
a Triangle
Math On the Spot
Sum of Angle Measures of a Triangle
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The sum of the measures of the
angles in a triangle is 180°.
2
1
3
m∠1 + m∠2 + m∠3 = 180°
EXAMPLE 1
6.8.A
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.
The sum of the angle measures
m∠1 + m∠2 + m∠3 = 180° in a triangle is 180°.
x
65° + 65° + x = 180°
Write an equation.
130° + x = 180°
−130°
65°
Add.
−130°
65°
Subtract 130° from both
sides.
x = 50°
The angle at the top of the triangle measures 50°.
Math Talk
Mathematical Processes
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Can a triangle have two
obtuse angles? Why
or why not?
YOUR TURN
Find the unknown angle measures.
2.
D
100°
x
E
K
3.
x
55°
F
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and Intervention
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430
Unit 5
J
x=
x=
71°
56°
L
Finding Angles in an
Equilateral Triangle
Recall that an equilateral triangle has three congruent sides and three
congruent angles.
Math On the Spot
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EXAMPL 2
EXAMPLE
6.8.A
Find the angle measures in the equilateral triangle.
3x = 180°
Write an equation.
3x = ____
180°
___
Divide both sides by 3.
3
3
x = 60°
x
x
x
Each angle in an equilateral triangle measures 60°.
Reflect
4. Multiple Representations Write a different equation to find the angle
measures in Example 2. Will the answer be the same? Explain.
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5. Draw Conclusions Triangle ABC is a right triangle. What conclusions
can you draw about the measures of the angles of the triangle?
YOUR TURN
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°.
8. The measures of two of the angles are 35°.
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and Intervention
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Lesson 15.2
431
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 =
+
S
105°
+x=
T
x
+x=
-
42°
-
R
x=
K
3.
A
x
4.
96°
x
42°
L
28°
C
x=
x=
5. G
M
6.
61°
33°
F
x
28°
H
x=
P x
8. The measures of two of the angles are 50° and 30°.
ESSENTIAL QUESTION CHECK-IN
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?
432
Unit 5
59° N
x=
7. The measures of two of the angles are 45°.
?
?
B
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J
Name
Class
Date
15.2 Independent Practice
Personal
Math Trainer
6.8.A
Figure ABCD represents
a garden crossed by
___
straight walkway AC. Use the figure for 10–15.
A
16. An observer at point O sees airplane
P directly over airport A. The observer
measures the angle of the plane at 40.5°.
B
P
57°
100°
D
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Assessment and
Intervention
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32°
88°
C
O
10. Find m∠DAC.
40.5°
A
Find m∠P.
11. Explain how you found m∠DAC.
The map shows the intersection of three
streets in San Antonio’s River Walk district.
Use the map for 17–18.
B
12. Find m∠BAC.
N
ll
S
48°
© Houghton Mifflin Harcourt Publishing Company
13. Explain how you found m∠BAC.
A
h
y’
n
C
17. Find the measures of the three angles of
the triangle.
14. Find m∠DAB.
15. Explain how you found m∠DAB.
18. Explain how you found the angle
measures.
Lesson 15.2
433
FOCUS ON HIGHER ORDER THINKING
Work Area
19. Persevere in Problem Solving Find the measure of ∠ACB. Explain how
you found your answer.
A
83°
x
D
148°
B
C
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?
B
A
C
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.
434
Unit 5
© Houghton Mifflin Harcourt Publishing Company
D
LESSON
15.3
?
Relationships
Between Sides and
Angles in a Triangle
ESSENTIAL QUESTION
EXPLORE ACTIVITY
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…
How can you use the relationships between side lengths
and angle measures in a triangle to solve problems?
6.8.A
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.
A Use geometry software to make triangle ABC.
Make ∠A the smallest angle.
B
C
A
B Choose one vertex and drag it so that you
lengthen the side of the triangle opposite angle
A. Describe what happens to ∠A.
B
C
© Houghton Mifflin Harcourt Publishing Company
A
C Drag the vertex to shorten the side opposite ∠B.
What happens to ∠B?
D 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.
B
A
C
Lesson 15.3
435
Using the Relationship Between Sides
and Angles in a Triangle
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EXAMPLE 1
My Notes
6.8.A
A 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.
B
100°
A
50°
30°
AC = 9 cm
The longest side is opposite the largest angle.
AB = 4.5 cm
The shortest side is opposite the smallest angle.
BC = 7 cm
The midsize side is opposite the midsize angle.
B 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.
B
20
13
m∠C = 40°
A
The largest angle is opposite the
17.6
longest side.
The smallest angle is opposite the shortest side.
m∠B = 60°
The midsize angle is opposite the midsize side.
m∠A = 80°
YOUR TURN
1.
2.
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and Intervention
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436
Unit 5
AC =
BC =
A
88°
57°
35°
m∠B =
6
m∠C =
A
C
C
Triangle ABC has angle measures of 45°, 58°, and 77°.
Match each angle with its correct measure.
m∠A =
C
B
Triangle ABC has side lengths of 11, 16, and 19.
Match each side with its correct length.
AB =
C
5
7
B
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Math On the Spot
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.
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.
EXAMPL 2
EXAMPLE
Problem
Solving
FPO
Math On the Spot
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6.8.A
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.
© Houghton Mifflin Harcourt Publishing Company
Justify and Evaluate
Solve
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.
45°
90°
45°
90°
45°
longest side
45°
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.
Lesson 15.3
437
YOUR TURN
3.
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Math Trainer
Online Assessment
and Intervention
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?
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Guided Practice
B
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°
105°
m∠
A
= 60°
3. Vocabulary Explain how the relationship between
the sides and angles of a triangle applies to equilateral
triangles. (Example 2)
43°
32°
C
N
58 in.
20 in.
M
51 in.
P
?
?
ESSENTIAL QUESTION CHECK-IN
5. Describe the relationship between the lengths of the sides and the
measures of the angles in a triangle.
438
Unit 5
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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)
Name
Class
Date
15.3 Independent Practice
Personal
Math Trainer
6.8.A
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Use the figure for 6–8.
Online
Assessment and
Intervention
The figure shows the angle measurements
formed by two fenced-in animal pens that
share a side. Use the figure for 9–10.
G
B
A
57°
58°
58°
F
58.5°
61°
H
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.
68°
54°
D
65°
C
_
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.
© Houghton Mifflin Harcourt Publishing Company
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?
10. What is the longest segment of fencing in
△ABC? Explain your reasoning.
8. Which side of the triangle is the longest?
Explain how you found the answer.
11. Find the longest segment of fencing in the
figure. Explain your reasoning.
Lesson 15.3
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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.
FOCUS ON HIGHER ORDER THINKING
Work Area
Z
13. Persevere in Problem Solving
Determine the shortest line segment in the
figure. Explain how you found the answer.
30°
30°
X
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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Unit 5
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MODULE QUIZ
Ready
Personal
Math Trainer
15.1 Determining When Three Lengths Form a Triangle
Online Assessment
and Intervention
Determine whether the three side lengths form a triangle.
my.hrw.com
1. 3, 5, 7
2. 9, 15, 4
3. 17, 5, 23
4. 28, 16, 38
15.2 Sum of Angle Measures in a Triangle
Find the unknown angle measures.
5.
6.
88°
41°
38°
112°
15.3 Relationships Between Sides and Angles in a Triangle
Match each of the given measures to the correct side or angle.
7. 11, 7.5, 13
8. 24°, 44°, 112°
A
89°
16
© Houghton Mifflin Harcourt Publishing Company
D
35°
C
7
12
56°
B
E
F
ESSENTIAL QUESTION
9. How can you describe the relationships among angles and sides in
a triangle?
Module 15
441
Personal
Math Trainer
MODULE 15 MIXED REVIEW
Texas Test Prep
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
B 6
D 19
my.hrw.com
5. Which of these could be the value of x in
the triangle below?
A
50°
11
29
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?
4x
43°
C
87°
22
B
A 5
B 6
A the 3-foot bar
C
B the 5-foot bar
C
Online
Assessment and
Intervention
7
D 10
the 7-foot bar
D the 12-foot bar
3. What is the measure of the missing angle
in the triangle below?
Gridded Response
6. Find m∠Z.
X
F
133°
29°
Y
.
D
65°
56°
A 39°
C
B 49°
D 69°
E
59°
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?
442
A 39.2°
C
B 52.8°
D 127.2°
Unit 5
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