5.4 Inequalities in one triangle

Slide 5-1
Copyright © 2014 Pearson Education, Inc.

Objectives

1.

Learn the Triangle Relationship Between Length of a Side
and Size of Its Opposite Angle.

2.

Use the Triangle Inequality Theorem.

5.4 Inequalities in One Triangle

Draw Three different triangles. Scalene, Isosceles, and Equilateral

Some text here about the topic of discussion

Slide 5-2

Theorem 5.5-3 Triangle Inequality Theorem

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Multiple Choice

Identify the smallest angle.

• ∠A

• ∠B

• ∠C

Multiple Choice

Identify the largest angle.

• ∠N

• ∠Q

• ∠P

Multiple Choice

Identify the smallest angle.

• ∠K

• ∠L

• ∠M

Slide 5-3

Theorem 5.5-4 Converse of the Triangle

Inequality Theorem

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Multiple Choice

Identify the shortest side measure.

• QP

• QM

• MP

Multiple Choice

List the side measures in order, shortest to longest.

• QM, QP, MP

• MQ, MP, QP

• MP, MQ, QP

Multiple Choice

Identify the longest side measure.

• AC

• BA

• BC

Slide 5-4

ExampleTriangle—Side Length vs. Angle

Measure

A town park is triangular. A landscape architect
wants to place a bench at the corner with the largest
angle. Which two streets form the corner with the
largest angle?

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Slide 5-5

ExampleTriangle—Side Length vs. Angle

Measure

A town park is triangular. A landscape architect
wants to place a bench at the corner with the largest
angle. Which two streets form the corner with the
largest angle?

Solution
Hollingsworth Road is the longest street, so it is
opposite the largest angle. MLK Boulevard and
Valley Road form the largest angle.

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Slide 5-6

Theorem 5.5-5 Triangle Inequality Theorem

for Sum of Lengths of Sides

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Slide 5-7

ExampleUsing the Triangle Inequality

Theorem (Theorem 5.5-5)

Can a triangle have sides with the given lengths?
Explain.
a. 3 ft, 7 ft, 8 ft

b. 5 ft, 10 ft, 15 ft

Solution
a.

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

Copyright © 2014 Pearson Education, Inc.

Slide 5-8

ExampleUsing the Triangle Inequality

Theorem (Theorem 5.5-5)

Can a triangle have sides with the given lengths?
Explain.
a. 3 ft, 7 ft, 8 ft

b. 5 ft, 10 ft, 15 ft

Solution
b.

No. The sum of 5 and 10 is not greater than 15. This
contradicts the theorem.

Copyright © 2014 Pearson Education, Inc.

Multiple Choice

Can the following side measures form a triangle?
  3, 5, 8 

• Yes

• No

Slide 5-9

ExampleFinding Possible Side Lengths

Two sides of a triangle are 5 ft and 8 ft long. What is
the range of possible lengths for the third side?
Solution
Let x represent the length of the third side. Use the
Triangle Inequality Theorem for Sum of Lengths of
Sides to write three inequalities. Then solve each
inequality for x.

Copyright © 2014 Pearson Education, Inc.

Slide 5-10

ExampleFinding Possible Side Lengths

Two sides of a triangle are 5 ft and 8 ft long. What is
the range of possible lengths for the third side?
Solution
Numbers that satisfy x > 3 and x > –3 must be
greater than 3. So, the third side must be greater than
3 ft and less than 13 ft long.

Copyright © 2014 Pearson Education, Inc.

Multiple Choice

What is the range of values of x for a triangle's third side given side measures of 8 and 15?

• between 8 and 15

• between 7 and 23

• less than 7, greater than 23

Multiple Choice

What is the range of values of x for a triangle's third side given side measures of 4 and 12?

• between 8 and 16

• less than 4, greater than 12

• between 4 and 12