LESSON
7.3
Name
Triangle Inequalities
Class
7.3
Date
Triangle Inequalities
Essential Question: How can you use inequalities to describe the relationships among side
lengths and angle measures in a triangle?
Texas Math Standards
The student is expected to:
G.5.D
Explore
Verify the Triangle Inequality theorem using constructions and apply the
theorem to solve problems.
Exploring Triangle Inequalities
A triangle can have sides of different lengths, but are there limits to the lengths of any of the sides?
In this activity, you will look at what limits exist for the third side of a triangle, once the first two
sides have been defined. You will construct a △ABC where you know two side lengths,
_
AB = 4 inches and BC = 2 inches, and look at what limits exist for the length of AC,
the third side of the triangle.
Mathematical Processes
G.1.C
Select tools, including real objects, manipulatives, paper and pencil, and
technology as appropriate, and techniques, including mental math,
estimation, and number sense as appropriate, to solve problems.

Language Objective
1.A, 2.C.3, 2.C.4, 4.F, 4.G
Consider a △ABC where you know two side
lengths, AB = 4 inches and BC = 2 inches.
Draw your construction on a separate piece of
paper. Draw a point A. Open up your compass to
the length 4 inches and make an arc centered at A.
Draw a point on the arc and label it B.
Explain to a partner how to show the three inequalities generated for a
triangle with side lengths a, b, and c.
ENGAGE
The sum of any two side lengths of a triangle will be
greater than the length of the third side. If the sides
of a triangle are not congruent, then the largest
angle will be opposite the longest side and the
smallest angle will be opposite the shortest side.

© Houghton Mifflin Harcourt Publishing Company
Essential Question: How can you use
inequalities to describe the
relationships among side lengths and
angle measures in a triangle?
Resource
Locker
G.5.D Verify the Triangle Inequality theorem using constructions and apply the theorem to
solve problems.
A
B
A
B
_
Next, measure out BC with the compass so that
BC = 2 inches. Place the compass point at B and
draw a circle with radius BC.
The circle shows all the possible locations for the
remaining vertex of the triangle.
PREVIEW: LESSON
PERFORMANCE TASK
View the Engage section online. Discuss the photo,
making sure that students understand the objective of
an orienteering competition and the tools used by
competing teams. Then preview the Lesson
Performance Task.
Module 7
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Lesson 3
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© Houghto
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C
Choose and label a final vertex point C so it is
located on the circle. Using a straightedge, draw the
segments to form a triangle.
C
EXPLORE
Are there any places on the circle where point C
cannot lie? Explain.
Exploring Triangle Inequalities
Point C cannot lie on the two points of the
→
‾ because then the
circle that intersect AB
A
B
INTEGRATE TECHNOLOGY
sides will overlap to form a straight line,
Students have the option of completing the triangle
inequality activity either in the book or online.
not a triangle.
D
Use a ruler to measure the lengths of the
three sides of your triangle and record
the lengths in a table.
Length
¯
of AB
QUESTIONING STRATEGIES
Triangle
ABC
Possible answer: 4 in., 2 in., 3.2 in.
E
Length
_ of Length
_ of
BC
AC
How do you decide if three lengths can be the
side lengths of a triangle? Check the sum of
each pair of two sides. The sum must be greater
than the third side.
The figures below show two
_ other examples of △ABC that could have
_ been formed.
What are the values that AC approaches when point C approaches AB?
C
A
C
B
A
B
AC approaches 2 inches or 6 inches.
Reflect
Use the side lengths from your table to make the following comparisons. What do
you notice?
AB + BC ? AC
BC + AC ? AB
AC + AB ? BC
The sum of any of the two sides is greater than the third side.
2.
Measure the angles of some triangles with a protractor. Where is the smallest angle
in relation to the shortest side? Where is the largest angle in relation to
the longest side?
The smallest angle is opposite the shortest side; the largest angle is opposite the longest
side.
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© Houghton Mifflin Harcourt Publishing Company
1.
Lesson 3
PROFESSIONAL DEVELOPMENT
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Learning Progressions
23/02/14 4:04 PM
In this lesson, students add to their knowledge of triangles by using a variety of
tools to verify the Triangle Inequality Theorem. Students also explore how to
order the side lengths given the angle measures of the triangle and how to predict
the possible lengths of the third side of a triangle, given the lengths of two of the
sides. Triangles have important uses in everyday life and in future mathematics
study, including trigonometry. All students should develop fluency with the
properties of triangles as they continue their study of geometry.
Triangle Inequalities 382
Discussion How does your answer to the previous question relate to isosceles triangles or equilateral
triangles?
When angles in a triangle have the same measure, the sides opposite those angles also
3.
EXPLAIN 1
have the same measure.
Using the Triangle Inequality
Theorem
Explain 1
Using the Triangle Inequality Theorem
The Explore shows that the sum of the lengths of any two sides of a triangle is greater than the length of the third
side. This can be summarized in the following theorem.
AVOID COMMON ERRORS
Triangle Inequality Theorem
Some students may have difficulty understanding
why all three inequalities must be checked for the
Triangle Inequality Theorem. One example may be
side lengths of 5 cm, 5 cm, and 10 cm. Straws with
these lengths look like they will make a triangle, but
they do not. Have them do several examples with
different side lengths to test the theorem.
A The sum of any two side lengths of a triangle is greater than the third side length.
A
C
B
C
B
AB + BC > AC
AB + BC > AC
BC + AC > AB
BC + AC > AB
AC + AB > BC
AC + AB > BC
To be able to form a triangle, each of the three inequalities must be true. So, given three side lengths, you can test to
determine if they can be used as segments to form a triangle. To show that three lengths cannot be the side lengths of
a triangle, you only need to show that one of the three triangle inequalities is false.
Example 1
QUESTIONING STRATEGIES
For side lengths a, b, and c of a triangle, how
many inequalities must be true? Write
them. 3; a < b + c, b < a + c, c < a + b

Use the Triangle Inequality Theorem to tell whether a triangle can have
sides with the given lengths. Explain.
4, 8, 10
?
4 + 8 > 10
?
4 + 10 > 8
12 > 10 ✓
?
8 + 10 > 4
14 > 8 ✓
18 > 4 ✓
© Houghton Mifflin Harcourt Publishing Company
Conclusion: The sum of each pair of side lengths is greater than the third length. So, a triangle can have
side lengths of 4, 8, and 10.

7, 9, 18
?
?
7 + 9 > 18
16 > 18
7 + 18 >
x
?
9 + 18 > 7
9
25 > 9
✓
27 > 7
✓
Conclusion:
Not all three inequalities are true. So, a triangle cannot have these three side lengths.
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Lesson 3
COLLABORATIVE LEARNING
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Small Group Activity
Give all students in groups pieces of raw spaghetti. Have each student break three
pieces of spaghetti in different lengths and measure the length of each piece. Then
have a group member write the three inequalities for those lengths. Have another
group member analyze the inequalities and conjecture if the three lengths will
form a triangle. Ask the fourth student to position the pieces to show a triangle or
to show no triangle. Switch roles and repeat the activity.
383
Lesson 7.3
2/20/14 10:57 PM
Reflect
4.
Can an isosceles triangle have these side lengths? Explain. 5, 5, 10
No; These numbers do not result in three true inequalities.
5 + 5 ≯ 10, so no triangle can be drawn with these side lengths.
5.
How do you know that the Triangle Inequality Theorem applies to all equilateral triangles?
Since all sides are congruent, the sum of any two side lengths will be greater than the third
side.
Your Turn
Determine if a triangle can be formed with the given side lengths. Explain your
reasoning.
6.
12 units, 4 units, 17 units
No; 12 + 4 ≯ 17
Explain 2
7.
24 cm, 8 cm, 30 cm
Yes; 24 + 8 > 30, 8 + 30 > 24,
and 24 + 30 > 8
Finding Possible Side Lengths in a Triangle
From the Explore, you have seen that if given two side
lengths for a triangle, there are an infinite number of
side lengths available for the third side. But the third side
is also restricted to values determined by the Triangle
Inequality Theorem.
b
a
© Houghton Mifflin Harcourt Publishing Company
Represent the unknown side length using a variable x.
x
b
a
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Triangle Inequalities 384
Here is an example with a = 5 and b = 6.
EXPLAIN 2
x
5
6
Finding Possible Side Lengths in a
Triangle
You can use the Triangle Inequality Theorem to find possible values for x.
x+5>6
x+6>5
x>1
CONNECT VOCABULARY
5+6>x
x > -1
11 > x
Ignore the inequality with a negative value, since a triangle cannot have a negative side length. Combine the other two
inequalities to find the possible values for x.
Relate the range of values for a third side length of a
triangle given two side lengths by stating that the
third side length must be greater than the difference
of the other two side lengths and also less than the
sum of the other two side lengths.
1 < x < 11
Example 2
Find the range of values for x using the Triangle Inequality Theorem.

12
x
10
x + 10 > 12
x> 2
x + 12 > 10
x > -2
10 + 12 > x
22 > x
2 < x < 22

15
x
© Houghton Mifflin Harcourt Publishing Company
15
x + 15 > 15
x + 15 > 15
15 + 15 > x
x > 0
x > 0
30 > x
0 < x < 30
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Lesson 3
LANGUAGE SUPPORT
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Visual Cues
Help students understand how to apply the inequality relationships in this lesson
by suggesting they list all the angles and sides before doing an example or exercise.
Then, they can list the angles in increasing order and write the side lengths
opposite those angles in the same order.
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Lesson 7.3
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Reflect
8.
QUESTIONING STRATEGIES
Discussion Suppose you know that the length of the base of an isosceles triangle is 10,
but you do not know the lengths of its legs. How could you use the Triangle Inequality
Theorem to find the range of possible lengths for each leg? Explain.
Possible answer: If x represents the length of one leg, then by the Triangle Inequality
How do find the range for the length of the
third side of a triangle? The range is r < x < s,
where r is the difference of the two given side
lengths and s is the sum of the two given side
lengths.
Theorem, solve for x + x > 10 and x + 10 > x. The solution of the first inequality is x > 5.
The solution of the second inequality is 10 > 0, which is always true. So the range of
possible lengths for each leg is x > 5.
How do you interpret the compound
inequality a < x < b as individual
inequalities? a < x and x < b
Your Turn
Find the range of values for x using the Triangle Inequality Theorem.
9.
21
14
10.
x
9
x
EXPLAIN 3
18
x + 14 > 21
x>7
x + 21 > 14
21 + 14 > x
35 > x
Explain 3
Ordering a Triangle’s Angle Measures Given
Its Side Lengths
x > -7
x + 9 > 18
x>9
x + 18 > 9
7 < x < 35
18 + 9 > x
27 > x
x > -9
Ordering a Triangle’s Angle Measures
Given Its Side Lengths
9 < x < 27
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Technology
From the Explore Step D, you can see that as side AC approaches its shortest possible length, the
angle opposite that side approaches 0°. Likewise, as side AC approaches its longest possible
length, the angle opposite that side approaches 180°.
A
A
C
C
B
B
A
As side
As side
AC gets
AC gets
shorter,
shorter,
m∠Bmapproaches
∠B approaches
0° 0°
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A
B
C
B
As side
As side
AC gets
AC gets
longer,
longer,
m∠Bmapproaches
∠B approaches
180°180°
386
© Houghton Mifflin Harcourt Publishing Company
C
Have students use geometry software to create a
scalene triangle. Ask them to use the measuring
features to measure the side lengths. Then have them
measure each angle and verify that the largest angle is
opposite the longest side length and that the smallest
angle is opposite the shortest side length. Ask them to
drag the vertices to vary the side lengths and then
observe that the angle measures are ordered in the
same way as in the original triangle.
Lesson 3
22/01/15 9:54 AM
Triangle Inequalities 386
Side-Angle Relationships in Triangles
QUESTIONING STRATEGIES
If two sides of a triangle are not congruent, then the larger angle is opposite the longer side.
How do you know the greatest angle is
opposite the longest side in a triangle? If one
side of a triangle is longer than another, then the
angle opposite the longer side is larger than the
angle opposite the shorter side.
C
A
AC > BC
m∠B > m∠A
B
Example 3
For each triangle, order its angle measures from least to greatest.
C

Longest side length: AC
Greatest angle measure: m∠B
15
12
Shortest side length: AB
Least angle measure: m∠C
Order of angle measures from least to greatest: m∠C, m∠A, m∠B
A
B
10

BC
Longest side length:
Greatest angle measure:
C
m∠A
Shortest side length:
AB
Least angle measure:
m∠C
21
A
22
20
B
Order of angle measures from least to greatest: m∠C, m∠B, m∠A
© Houghton Mifflin Harcourt Publishing Company
Your Turn
For each triangle, order its angle measures from least to greatest.
11.
Longest side length: AB
m∠A, m∠B, m∠C
32
C
12. C
7
A
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Lesson 7.3
A
Shortest side length: CB
15
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40
B
Longest side length: BC
25
24
Shortest side length: AC
B
m∠B, m∠C, m∠A
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Lesson 3
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Ordering a Triangle’s Side Lengths Given
Its Angle Measures
Explain 4
EXPLAIN 4
From the Explore Step D, you can see that as the m∠B approaches 0°, the side opposite that angle approaches its
shortest possible length. Likewise, as m∠B approaches 180°, the side opposite that angle approaches its greatest
possible length.
C
A
C
C
B
A
A
B
B
A
Ordering a Triangle’s Side Lengths
Given Its Angle Measures
C
B
AVOID COMMON ERRORS
As m∠B
approaches
0°, side
AC gets
shorter
As m∠B
approaches
0°, side
AC gets
shorter
As m∠B
approaches
180°,180°,
sideside
AC gets
longer
As m∠B
approaches
AC gets
longer
Some students may think that they can use angle
measures to compare the side lengths of different
triangles. Explain that angle measures can be used to
order the side lengths only within a single triangle.
Give an example of why this must be the case, such as
a very small triangle with an obtuse angle and a very
large equilateral triangle. The obtuse angle is greater
than the 60° angle of the equilateral triangle, but its
opposite side may be shorter.
Angle-Side Relationships in Triangles
If two angles of a triangle are not congruent, then the longer side is opposite the larger angle.
C
B
A
Example 4
m∠B > m∠A
AC > BC
For each triangle, order the side lengths from least to greatest.

QUESTIONING STRATEGIES
C
Greatest angle measure: m∠B
50°
Longest side length: AC
Least angle measure: m∠A
A
How do you order the side lengths of a
triangle given the angle measures?
Explain. The side lengths will be in the same order
as the measure of the angles opposite the side
lengths. For example, the greatest side length is
opposite the greatest angle measure. Use the rule
that if one angle of a triangle is larger than another,
then the side opposite the larger angle is longer
than the side opposite the smaller angle.
100° B
30°
Shortest side length: BC

Greatest angle measure:
m∠A
Longest side length:
BC
Least angle measure:
m∠C
Shortest side length:
AB
Order of side lengths from least to great:
Module 7
C
45°
A 70°
65° B
AB, AC, BC
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© Houghton Mifflin Harcourt Publishing Company
Order of side lengths from least to greatest: BC, AB, AC
Lesson 3
DIFFERENTIATE INSTRUCTION
GE_MTXESE353886_U2M07L3.indd 388
Manipulatives
2/20/14 10:55 PM
Give students a number of straws and ask them to cut them into various lengths.
Then have them measure each length. Ask them to make a table listing the
measures of all possible combinations of the three lengths. Then have them
manipulate the straws to see if they can form a triangle. Highlight sets of three
measurements that do not form a triangle.
Triangle Inequalities 388