5-5 Inequalities in Triangles

5-5

Inequalities in Triangles

What You'll Learn

? To use inequalities involving

angles of triangles

? To use inequalities involving

sides of triangles

. . . And Why

To locate the largest corners on a triangular backyard deck, as in Example 2

Check Skills You'll Need

GO for Help Lessons 1-8 and 5-4

Graph the triangles with the given vertices. List the sides in order from shortest to longest. 1?4. See back of book.

1. A(5, 0), B(0, 8), C(0, 0)

2. P(2, 4), Q(-5, 1), R(0, 0)

3. G(3, 0), H(4, 3), J(8, 0)

4. X(-4, 3), Y(-1, 1), Z(-1, 4)

Recall the steps for indirect proof. 5. You want to prove m&A . m&B. Assume that mlA K mlB.

Write the first step of an indirect proof. 6. In an indirect proof, you deduce that AB $ AC is false. What conclusion

can you make? AB R AC

5-5

1. Plan

Objectives

1 To use inequalities involving angles of triangles

2 To use inequalities involving sides of triangles

Examples

1 Applying the Corollary 2 Real-World Connection 3 Using Theorem 5-11 4 Using the Triangle Inequality

Theorem 5 Finding Possible Side Lengths

1 Inequalities Involving Angles of Triangles

When you empty a container of juice into two glasses, it is difficult to be sure that the glasses get equal amounts. You can be sure, however, that each glass holds less than the original amount in the container. This is a simple application of the Comparison Property of Inequality.

Key Concepts

Property

Comparison Property of Inequality

If a = b + c and c . 0, then a . b.

Proof

Proof of the Comparison Property

Given: a = b + c, c . 0 Prove: a . b

Statements

Reasons

1. c . 0 2. b + c . b + 0 3. b + c . b 4. a = b + c 5. a . b

1. Given 2. Addition Property of Inequality 3. Simplify. 4. Given 5. Substitute a for b + c in Statement 3.

The Comparison Property of Inequality allows you to prove the following corollary to the Exterior Angle Theorem for triangles (Theorem 3-13).

Lesson 5-5 Inequalities in Triangles 289

Math Background

Theorems 5-10 and 5-11 can be treated as extending the Isosceles Triangle Theorem and its converse to the case of inequality. These theorems enable students to prove in Exercise 41 that the shortest segment from a point to a line is perpendicular to the line.

More Math Background: p. 256D

Lesson Planning and Resources

See p. 256E for a list of the resources that support this lesson.

PowerPoint

Bell Ringer Practice

Check Skills You'll Need For intervention, direct students to:

Finding Distance Lesson 1-8: Example 1 Extra Skills, Word Problems, Proof

Practice, Ch. 1

Indirect Proof Lesson 5-4: Examples 3, 5 Extra Skills, Word Problems, Proof

Practice, Ch. 5

Special Needs L1 To illustrate that Theorem 4-10 and 4-11 only apply to one triangle, draw TUV along with a similar but smaller triangle, ABC. Show that TV AB does not imply that TV AC.

learning style: visual

Below Level L2 Using geometry software to alter the sides and angles of triangles (beginning with an isosceles triangle) may help students understand Theorems 5-10 and 5-11.

learning style: visual

289

2. Teach

Guided Instruction

1 EXAMPLE

Remind students that a corollary is both a statement that follows directly from a theorem and a theorem itself.

2 EXAMPLE Tactile Learners

Have students construct a triangle to model the problem, using sides 18 cm, 21 cm, and 27 cm long to see that the larger angles are opposite the longer sides.

PowerPoint

Additional Examples

1 Explain why m&4 m&5. B 5 Y 2

41

A

X

3C

ml4 S ml2 by the Corollary to the Exterior Angle Theorem,

ml2 ml5 because l2 and l5 are congruent corresponding angles, and

ml4 S ml5 by substitution.

2 In RGY, RG = 14, GY = 12, and RY = 20. List the angles from largest to smallest. lG, lY, lR

Key Concepts

Corollary

Corollary to the Triangle Exterior Angle Theorem

The measure of an exterior angle of a triangle is greater than the measure of each of its remote interior angles.

m&1 . m&2 and m&1 . m&3

3

2

1

Proof

Proof of the Corollary

Given: &1 is an exterior angle of the triangle. Prove: m&1 . m&2 and m&1 . m&3. Proof: By the Exterior Angle Theorem, m&1 = m&2 + m&3. Since m&2 . 0

and m&3 . 0, you can apply the Comparison Property of Inequality and conclude that m&1 . m&2 and m&1 . m&3.

1 EXAMPLE Applying the Corollary

In the diagram, m&2 = m&1 by the Isosceles Triangle Theorem. Explain why m&2 . m&3.

O

3

P

By the corollary to the Exterior Angle Theorem,

1

m&1 . m&3. So, m&2 . m&3 by substitution.

4

Quick Check 1 Explain why m&OTY . m&3. mlOTY S ml2

2 T

Y

by the Comparison Prop. of Ineq. Since it was

proven that ml2 S ml3, then by the Trans. Prop. mlOTY S ml3. You will prove the following inequality theorem in the exercises.

Key Concepts

Theorem 5-10

If two sides of a triangle are not congruent, then

Y

the larger angle lies opposite the longer side.

If XZ . XY, then m&Y . m&Z.

X

Z

2 EXAMPLE Real-World Connection

Real-World Connection

Careers Landscape architects blend structures with decorative plantings.

Deck Design A landscape architect is designing a triangular deck. She wants to place benches in the two larger corners. Which corners have the larger angles?

Corners B and C have the larger angles. They are opposite the two longer sides of 27 ft and 21 ft.

Quick Check 2 List the angles of #ABC in order from

smallest to largest. lA, lC, lB

A 21 ft

27 ft C

18 ft B

290

290 Chapter 5 Relationships Within Triangles

Advanced Learners L4 Have students find the range of possible values for the length of the third side of a triangle whose other side lengths are a and b. If bL a, b?a R c R a?b.

learning style: verbal

English Language Learners ELL Use the proof of the corollary before Example 1 to review the meaning of the term corollary. A corollary is a statement that can be proved easily by applying the theorem.

learning style: verbal

12 Inequalities Involving Sides of Triangles

Theorem 5-10 on the preceding page states that the larger angle is opposite the longer side. The converse is also true.

Key Concepts

Theorem 5-11

If two angles of a triangle are not congruent,

B

then the longer side lies opposite the larger angle. C

If m&A . m&B, then BC . AC.

A

Proof

Indirect Proof of Theorem 5-11

Given: m&A . m&B Prove: BC . AC

Step 1 Assume BC AC. That is, assume BC , AC or BC = AC.

Step 2 If BC , AC, then m&A , m&B (Theorem 5-10). This contradicts the given fact that m&A . m&B. Therefore, BC , AC must be false.

If BC = AC, then m&A = m&B (Isosceles Triangle Theorem). This also contradicts m&A . m&B. Therefore, BC = AC must be false.

Step 3 The assumption BC AC is false, so BC . AC.

D

E

C

1 A

B

B 2 A

3 A

B

B 4 A

5 A

B

E D C

D E C

E D C

E D C

E D C

B

Test-Taking Tip

Don't be distracted! Choice B lists the sides in order, but from longest to shortest, not shortest to longest.

Quick Check

3 EXAMPLE Using Theorem 5-11

Multiple Choice Which choice shows the sides of

T

#TUV in order from shortest to longest?

TV, UV, UT

UT, UV, TV

UV, UT, TV

TV, UT, UV

U 58 62 V

By the Triangle Angle-Sum Theorem, m&T 60.

58 60 62, so m&U m&T m&V. By Theorem 5-11, TV UV UT. The correct choice is A.

Y

3 List the sides of the #XYZ in order from shortest to longest. Explain your listing. YZ R XY R XZ since mlY 80.

X 40 60 Z

The lengths of three segments must be related in a certain way to form a triangle.

3 cm

3 cm

2 cm

2 cm

5 cm 3 cm, 3 cm, 5 cm

6 cm 2 cm, 2 cm, 6 cm

Notice that only one of the sets of three segments above can form a triangle. The sum of the smallest two lengths must be greater than the greatest length. This is Theorem 5-12 (see next page). You will prove it in the exercises.

Lesson 5-5 Inequalities in Triangles 291

Guided Instruction

3 EXAMPLE Error Prevention

Emphasize that Theorems 5-10 and 5-11 apply only within triangles, not between triangles.

4 EXAMPLE Math Tip

Ask: Why does comparing only the sum of the two shorter sides and the longest side tell whether a triangle can have the given lengths? If the least sum is greater than the greatest length, the other inequalities must be true also.

5 EXAMPLE

Discuss why "x 2 and x -2" can be written as x 2. Point out that the possible lengths are written as a compound inequality.

PowerPoint

Additional Examples

3 In ABC, &C is a right angle. Which is the longest side? AB

4 Can a triangle have sides with the given lengths? Explain. a. 2 cm, 2 cm, 4 cm no; 2 ? 2 4 b. 8 in., 15 in., 12 in. yes; 8 ? 12 S 15

5 In FGH, FG = 9 m and GH = 17 m. Describe the possible lengths of FH. 8 R FH R 26

Resources

? Daily Notetaking Guide 5-5 L3

? Daily Notetaking Guide 5-5--

Adapted Instruction

L1

Closure

Explain why each triangle below is impossible.

12 100? 15 18

12

30? 50?

25

32

In the first triangle, the side opposite the smallest angle is not the shortest side; the second triangle violates the Triangle Inequality Theorem.

291

3. Practice

Assignment Guide

1 A B 1-9, 30, 33

2 A B 10-29, 31, 32, 34-37

C Challenge

38-41

Test Prep Mixed Review

42-49 50-61

Homework Quick Check

To check students' understanding of key skills and concepts, go over Exercises 5, 22, 29, 30, 33.

Exercise 5 Ask: How do you know x 0? Side lengths must be positive numbers.

Exercise 6 Make sure that students can explain why &I is the largest angle in GHI.

Visual Learners

Exercises 7?9, 13?15 Have students draw and label each triangle to make sure that they identify opposite angles correctly.

GPS Guided Problem Solving

L3

Enrichment

L4

Reteaching

L2

Adapted Practice

L1

PraNcamte ice

Class

Date

L3

Practice 5-5

Inequalities in Triangles

Determine the two largest angles in each triangle.

1.

M 1 ft

N

5.5 ft 6 ft

L

2. 18 m

B

C 14 m

D 25 m

4. P 7 cm R

13 cm I

11 cm

5. 15 yd T

25 yd

A 20 yd

B

3.

Q

1.9 cm

S

4.1 cm

4.0 cm

R

6.

S

24 cm

39 cm K

A

55 cm

Can a triangle have sides with the given lengths? Explain.

7. 4 m, 7 m, and 8 m

8. 6 m, 10 m, and 17 m

10. 1 yd, 9 yd, and 9 yd

11. 11 m, 12 m, and 13 m

13. 1.2 cm, 2.6 cm, and 4.9 cm

14.

8

1 2

yd,

9

1 4

yd,

and

18

yd

List the sides of each triangle in order from shortest to longest.

16. B

47 C

17. L

41 A

56 O B

9. 4 in., 4 in., and 4 in. 12. 18 ft, 20 ft, and 40 ft 15. 2.5 m, 3.5 m, and 6 m

18.

S

75 T

107 R

List the angles of each triangle in order from largest to smallest.

19. S 1.7 D

20.

S

3.4

2.6

A

13 25

N 21 J

21. P

28 R

38

26

O

The lengths of two sides of a triangle are given. Describe the lengths possible for the third side.

22. 4 in., 7 in.

23. 9 cm, 17 cm

24. 5 ft, 5 ft

25. 11 m, 20 m

26. 6 km, 8 km

27. 24 in., 37 in.

? Pearson Education, Inc. All rights reserved.

292

Key Concepts

Theorem 5-12 Triangle Inequality Theorem

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

Y XY + YZ . XZ

YZ + ZX . YX ZX + XY . ZY

X

Z

4 EXAMPLE Using the Triangle Inequality Theorem

Can a triangle have sides with the given lengths? Explain.

a. 3 ft, 7 ft, 8 ft

b. 3 cm, 6 cm, 10 cm

For: Triangle Inequality Activity Use: Interactive Textbook, 5-5

3+7.8 8+7.3 3 + 8 . 7 Yes

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

3 + 6 10 No

The sum of 3 and 6 is not greater than 10, contradicting Theorem 5-12.

Quick Check

4 Can a triangle have sides with the given lengths? Explain.

a. 2 m, 7 m, and 9 m no; 2 ? 7 w 9

b. 4 yd, 6 yd, and 9 yd yes; 4 ? 6 S 9; 6 ? 9 S 4; and 4 ? 9 S 6

5 EXAMPLE Finding Possible Side Lengths

Quick Check

Algebra A triangle has sides of lengths 8 cm and 10 cm. Describe the lengths possible for the third side.

Let x represent the length of the third side. By the Triangle Inequality Theorem,

x + 8 . 10

x + 10 . 8

8 + 10 . x

x.2

x . -2

x , 18

The third side must be longer than 2 cm and shorter than 18 cm.

5 A triangle has sides of lengths 3 in. and 12 in. Describe the lengths possible for the third side. 9 R x R 15

EXEERxaCmIpSlEe 1S

For more exercises, see Extra Skill, Word Problem, and Proof Practice.

Practice and Problem Solving

A Practice by Example

Explain why ml1 S ml2. 1?3. See margin.

Example 1

1.

(page 290)

GO

for Help

2

2.

3

3.

32

2

1

4

1

4

1

43

292 Chapter 5 Relationships Within Triangles

1. l3 O l2 because they are vertical ' and ml1 S ml3 by Corollary to the Ext. l Thm. So, ml1 S ml2 by subst.

2. An ext. l of a k is larger than either remote int. l.

3. ml1 S ml4 by Corollary to the Ext. l Thm. and l4 O l2 because if n lines, then alt. int. ' are O.

16. No; 2 ? 3 w 6. 17. Yes; 11 ? 12 S 15;

12 ? 15 S 11; 11 ? 15 S 12.

Example 2 (page 290)

Example 3 (page 291)

Example 4 (page 292) Example 5 (page 292)

B Apply Your Skills

Exercise 29

18. No; 8 ? 10 w 19. 19. Yes; 1 ? 15 S 15;

15 ? 15 S 1. 20. Yes; 2 ? 9 S 10;

9 ? 10 S 2; 2 ? 10 S 9.

List the angles of each triangle in order from smallest to largest.

4.

K

4.3

2.7

L

5.8

lM, lL, lK

5. lD, lC, lE

M

C x 105

3x

E

D 6.

H

4

G

I

6

lG, lH, lI

7. #ABC, where AB = 8, 8. #DEF, where DE = 15, 9. #XYZ,where XY = 12,

BC = 5, and CA = 7 lA, lB, lC

EF = 18, and DF = 5 lE, lF, lD

YZ = 24, and ZX = 30 lZ, lX, lY

List the sides of each triangle in order from shortest to longest.

10.

O

11. G 28

12. T

45

M 75 N

MN, ON, MO

13. #ABC, with m&A = 90, m&B = 40, and m&C = 50 AC, AB, CB

110 H

F FH, GF, GH 14. #DEF, with

m&D = 20, m&E = 120, and m&F = 40 EF, DE, DF

30

U

V

TU, UV, TV

15. #XYZ, with

m&X = 51,

m&Y = 59, and

m&Z = 70 ZY, XZ, XY

Can a triangle have sides with the given lengths? Explain. 16?21. See margin.

16. 2 in., 3 in., 6 in.

17. 11 cm, 12 cm, 15 cm 18. 8 m, 10 m, 19 m

19. 1 cm, 15 cm, 15 cm

20. 2 yd, 9 yd, 10 yd

21. 4 m, 5 m, 9 m

x2 Algebra The lengths of two sides of a triangle are given. Describe the lengths possible for the third side.

22. 8 ft, 12 ft 4 R s R 20 23. 5 in., 16 in.11 R s R 21 24. 6 cm, 6 cm 0 R s R 12

25. 18 m, 23 m 5 R s R 41 26. 4 yd, 7 yd 3 R s R 11 27. 20 km, 35 km 15 R s R 5

28. Error Analysis The Shau family is crossing Kansas on Highway 70. A sign reads "Wichita 90 miles, Topeka 110 miles." Avi says, "I didn't know that it was only 20 miles from Wichita to Topeka." Explain to Avi why the distance between the two cities doesn't have to be 20 miles. See margin.

29. Writing Explain why the distance between the two peaks in the photograph is greater than the difference of the distances from the hiker to each of the peaks. See margin.

30. The Hinge Theorem The hypothesis of the Hinge Theorem is stated below. GPS The conclusion is missing. a?d. See margin.

Suppose two sides of one triangle are congruent to two sides of another triangle. If the included angle of the first triangle is larger than the included angle of the second triangle, then 9.

a. Draw a diagram to illustrate the hypothesis. b. The conclusion of the Hinge Theorem concerns the sides opposite the two

angles mentioned in the hypothesis. Write the conclusion. c. Draw a diagram to illustrate the converse. d. Converse of the Hinge Theorem Write the conclusion to this theorem.

Suppose two sides of one triangle are congruent to two sides of another triangle. If the third side of the first triangle is greater than the third side of the second triangle, then 9.

Lesson 5-5 Inequalities in Triangles 293

21. No; 4 ? 5 w 9.

28. Answers may vary. Sample: If Y is the distance between Wichita and Topeka, then 20 R Y R 200.

29. Let the distance between the peaks be d and the distances from the hiker to each of the peaks be a and b. Then d ? a S b and d ? b S a. Thus, d S b ? a and d S a ? b.

Careers Exercise 28 Traveling sales

representatives make a great effort to plan their routes efficiently. When traveling to multiple destinations over the course of a day, week, or month, a sales representative makes a schedule of routes that minimizes travel time and maximizes time with customers.

Exercise 29 Check that students use the Triangle Inequality Theorem in their explanation.

Technology Tip Exercise 30 Students can use

geometry software to investigate the Hinge Theorem and its converse.

30. a.

A

D

B

CE

F

b. The third side of the 1st k is longer than the third side of the 2nd k.

c. See diagram in part (a).

d. The included l of the first k is greater than the included l of the second k.

31. Answers may vary. Sample: The shortcut across the grass is shorter than the sum of the two paths.

293

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