Lesson 5 5 indirect proof and inequalities in one triangle

24 Chapter 5 Relationships Within Triangles

Objective To use inequalities involving angles and sides of triangles

In the Solve It, you explored triangles formed by various lengths of board. You may have noticed that changing the angle formed by two sides of the sandbox changes the length of the third side.

Essential Understanding Th e angles and sides of a triangle have special relationships that involve inequalities.

Property Comparison Property of Inequality

If a 5 b 1 c and c . 0, then a . b.

For a neighborhood improvement project, you volunteer to help build a new sandbox at the town playground. You have two boards that will make up two sides of the triangular sandbox. One is 5 ft long and the other is 8 ft long. Boards come in the lengths shown. Which boards can you use for the third side of the sandbox? Explain.

Inequalities in One Triangle

5-6

t t tt o lele f

Think about whether the shape of the triangle would be easy to play in.

Dynamic Activity Triangle Inequalities T

A C T I V I T I

E S T

AAAAAAAA C

A CC

I E SSSSSSSS

DY NAMIC

Proof of the Comparison Property of Inequality

Given: a 5 b 1 c, c . 0

Prove: a . b

Statements Reasons

1) c . 0 1) Given

2) b 1 c . b 1 0 2) Addition Property of Inequality

3) b 1 c . b 3) Identity Property of Addition

4) a 5 b 1 c 4) Given

5) a . b 5) Substitution

Proof

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Problem 1

Got It?

Lesson 5-6 Inequalities in One Triangle 325

Th e Comparison Property of Inequality allows you to prove the following corollary to the Triangle Exterior Angle Th eorem (Th eorem 3-11).

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 Triangle Exterior Angle Th eorem, m/1 5 m/2 1 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.

Applying the Corollary

Use the fi gure at the right. Why is ml2 S ml3?

In nACD, CB > CD, so by the Isosceles Triangle Th eorem, m/1 5 m/2. /1 is an exterior angle of nABD, so by the Corollary to the Triangle Exterior Angle Th eorem, m/1 . m/3. Th en m/2 . m/3 by substitution.

1. Why is m/5 . m/C?

You can use the corollary to Th eorem 3-11 to prove the following theorem.

Corollary Corollary to the Triangle Exterior Angle Theorem

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

If . . . /1 is an exterior angle

Then . . . m/1 . m/2 and m/1 . m/3

2 1

3

Proof

3

4

1

25

A

CD

B

Theorem 5-10

Theorem If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side.

If . . . XZ . XY

Then . . . m/Y . m/Z

You will prove Theorem 5-10 in Exercise 40.

X

Y

Z

G

U

I m C Th

G

How do you identify an exterior angle? An exterior angle must be formed by the extension of a side of the triangle. Here, /1 is an exterior angle of nABD, but /2 is not.

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Problem 2

Got It?

326 Chapter 5 Relationships Within Triangles

Using Theorem 5-10

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?

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

2. Suppose the landscape architect wants to place a drinking fountain at the corner with the second largest angle. Which two streets form the corner with the second-largest angle?

Th eorem 5-11 below is the converse of Th eorem 5-10. Th e proof of Th eorem 5-11 relies on indirect reasoning.

Indirect Proof of Theorem 5-11

Given: m/A . m/B

Prove: BC . AC

Step 1 Assume temporarily that BC 6 AC . Th at is, assume temporarily that either BC , AC or BC 5 AC .

Step 2 If BC , AC , then m/A , m/B (Th eorem 5-10). Th is contradicts the given fact that m/A . m/B. Th erefore, BC , AC must be false.

If BC 5 AC , then m/A 5 m/B (Isosceles Triangle Th eorem). Th is also contradicts m/A . m/B. Th erefore, BC 5 AC must be false.

Step 3 Th e temporary assumption BC 6 AC is false, so BC . AC .

120 yd

175 yd

105 yd

Hol ling

swo rth

Roa d

Va lle

y Ro

ad

MLK Boulevard

Theorem 5-11

Theorem If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle.

If . . . m/A . m/B

Then . . . BC . AC

B

A

C

Proof

G

A w l w

H o V

How do you fi nd the side opposite an angle? Choose an angle of a triangle. The side opposite it is the only side that does not have an endpoint at the vertex of the angle.

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Problem 3

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Problem 4

Got It?

Lesson 5-6 Inequalities in One Triangle 327

Using Theorem 5-11

Multiple Choice Which choice shows the sides of kTUV in order from shortest to longest?

TV, UV, UT UV, UT, TV

UT, UV, TV TV, UT, UV

By the Triangle Angle-Sum Th eorem, m/T 5 60. 58 , 60 , 62, so m/U , m/T , m/V. By Th eorem 5-11, TV , UV , UT. Choice A is correct.

3. Reasoning In the fi gure at the right, m/S 5 24 and m/O 5 130. Which side of nSOX is the shortest side? Explain your reasoning.

For three segments to form a triangle, their lengths must be related in a certain way. Notice that only one of the sets of segments below can form a triangle. Th e sum of the smallest two lengths must be greater than the greatest length.

Using the Triangle Inequality Theorem

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

A 3 ft, 7 ft, 8 ft B 5 ft, 10 ft, 15 ft

3 1 7 . 8 7 1 8 . 3 8 1 3 . 7 5 1 10 6 15

10 . 8 15 . 3 11 . 7 15 6 15

Yes. Th e sum of the lengths of any two No. Th e sum of 5 and 10 is not sides is greater than the length of the greater than 15. Th is contradicts third side. Th eorem 5-12.

4. Can a triangle have sides with the given lengths? Explain. a. 2 m, 6 m, and 9 m b. 4 yd, 6 yd, and 9 yd

T

U V 58 62

S

O

X

3 cm 2 cm2 cm3 cm

6 cm5 cm

Theorem 5-12 Triangle Inequality Theorem

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

XY 1 YZ . XZ YZ 1 XZ . XY XZ 1 XY . YZ

You will prove Theorem 5-12 in Exercise 45.

X

Y

Z

G

M s

B m

G

How do you use the angle measures to order the side lengths? List the angle measures in order from smallest to largest. Then replace the measure of each angle with the length of the side opposite.

C

A

How do you use the Triangle Inequality Theorem? Test each pair of side lengths. The sum of each pair must be greater than the third length.

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Problem 5

Got It?

328 Chapter 5 Relationships Within Triangles

Finding Possible Side Lengths

Algebra In the Solve It, you explored the possible dimensions of a triangular sandbox. Two of the sides are 5 ft and 8 ft long. What is the range of possible lengths for the third side?

Let x represent the length of the third side. Use the Triangle Inequality Th eorem to write three inequalities. Th en solve each inequality for x.

x 1 5 . 8 x 1 8 . 5 5 1 8 . x

x . 3 x . 23 x , 13

Numbers that satisfy x . 3 and x . 23 must be greater than 3. So, the third side must be greater than 3 ft and less than 13 ft.

5. A triangle has side lengths of 4 in. and 7 in. What is the range of possible lengths for the third side?

The lengths of two sides of the triangle are 5 ft and 8 ft.

The range of possible lengths of the third side

Use the Triangle Inequality Theorem to write three inequalities. Use the solutions of the inequalities to determine the greatest and least possible lengths.

Practice and Problem-Solving Exercises

Explain why ml1 S ml2.

6. 7. 8.

PracticeA See Problem 1.

2

3

1 4

2

1 34

3

1 4

2

Lesson Check Do you know HOW? Use kABC for Exercises 1 and 2.

1. Which side is the longest?

2. Which angle is the smallest?

3. Can a triangle have sides of lengths 4, 5, and 10? Explain.

Do you UNDERSTAND? 4. Error Analysis A friend tells you that she drew a

triangle with perimeter 16 and one side of length 8. How do you know she made an error in her drawing?

5. Reasoning Is it possible to draw a right triangle with an exterior angle measuring 88? Explain your reasoning.

A C

B

85

5

4

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Lesson 5-6 Inequalities in One Triangle 329

For Exercises 9–14, list the angles of each triangle in order from smallest to largest.

9. 10. 11.

12. nABC, where AB 5 8, 13. nDEF , where DE 5 15, 14. nXYZ , where XY 5 12, BC 5 5, and CA 5 7 EF 5 18, and DF 5 5 YZ 5 24, and ZX 5 30

For Exercises 15–20, list the sides of each triangle in order from shortest to longest.

15. 16. 17.

18. nABC , with 19. nDEF , with 20. nXYZ , with m/A 5 90, m/D 5 20, m/X 5 51, m/B 5 40, and m/E 5 120, and m/Y 5 59, and m/C 5 50 m/F 5 40 m/Z 5 70

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

21. 2 in., 3 in., 6 in. 22. 11 cm, 12 cm, 15 cm 23. 8 m, 10 m, 19 m

24. 1 cm, 15 cm, 15 cm 25. 2 yd, 9 yd, 10 yd 26. 4 m, 5 m, 9 m

Algebra Th e lengths of two sides of a triangle are given. Find the range of possible lengths for the third side.

27. 8 ft, 12 ft 28. 5 in., 16 in. 29. 6 cm, 6 cm

30. 18 m, 23 m 31. 4 yd, 7 yd 32. 20 km, 35 km

33. Think About a Plan You are setting up a study area where you will do your homework each evening. It is triangular with an entrance on one side. You want to put your computer in the corner with the largest angle and a bookshelf on the longest side. Where should you place your computer? On which side should you place the bookshelf? Explain.

• What type of triangle is shown in the fi gure? • Once you fi nd the largest angle of a triangle, how do you fi nd the

longest side?

34. Algebra Find the longest side of nABC , with m/A 5 70, m/B 5 2x 2 10, and m/C 5 3x 1 20.

See Problem 2.

5.8

2.7 4.3

L M

K

C

E

D

105 3x

x JG

4

6

H

See Problem 3.

O

M

N

75 45

G

F

H 28

110

T

U V 30

See Problem 4.

See Problem 5.

7 ft

EntranceApplyB

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330 Chapter 5 Relationships Within Triangles

35. Writing You and a friend compete in a scavenger hunt at a museum. Th e two of you walk from the Picasso exhibit to the Native American gallery along the dashed red line. When he sees that another team is ahead of you, your friend says, “Th ey must have cut through the courtyard.” Explain what your friend means.

36. Error Analysis Your family drives across Kansas on Interstate 70. A sign reads, “Wichita 90 mi, Topeka 110 mi.” Your little brother says, “I didn’t know that it was only 20 miles from Wichita to Topeka.” Explain why the distance between the two cities does not have to be 20 mi.

Reasoning Determine which segment is shortest in each diagram.

37. 38. 39.

40. Developing Proof Fill in the blanks for a proof of Th eorem 5-10: If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side.

Given: nTOY, with YO . YT

Prove: a. 9 . b. 9

Mark P on YO so that YP > YT. Draw TP.

Statements Reasons

1) YP > YT 1) Ruler Postulate

2) m/1 5 m/2 2) c. 9

3) m/OTY 5 m/4 1 m/2 3) d. 9

4) m/OTY . m/2 4) e. 9

5) m/OTY . m/1 5) f. 9

6) m/1 . m/3 6) g. 9

7) m/OTY . m/3 7) h. 9

41. Prove this corollary to Th eorem 5-11: Th e perpendicular segment from a point to a line is the shortest segment from the point to the line.

Given: PT ' TA

Prove: PA . PT

42. Probability A student has two straws. One is 6 cm long and the other is 9 cm long. She picks a third straw at random from a group of four straws whose lengths are 3 cm, 5 cm, 11 cm, and 15 cm. What is the probability that the straw she picks will allow her to form a triangle? Justify your answer.

Native American Gallery

Picasso Exhibit

SR

P

Q 30

40 C D

B A 114

110

32 30

X

W

Y

Z

48

47 95 40

3

4

1

2

O

YT

P

Proof

P

T A

ChallengeC

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Lesson 5-6 Inequalities in One Triangle 331

For Exercises 43 and 44, x and y are integers such that 1 R x R 5 and 2 R y R 9.

43. Th e sides of a triangle are 5 cm, x cm, and y cm. List all possible (x, y) pairs.

44. Probability What is the probability that you can draw an isosceles triangle that has sides 5 cm, x cm, and y cm, with x and y chosen at random?

45. Prove the Triangle Inequality Th eorem: Th e sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Given: nABC

Prove: AC 1 CB . AB

(Hint: On BC ) , mark a point D not on BC , so that DC 5 AC .

Draw DA and use Th eorem 5-11 with nABD.) A

C

B

Proof

Mixed Review

Write the fi rst step of an indirect proof of the given statement.

50. Th e side is at least 2 ft long. 51. nPQR has two congruent angles.

52. You know that AB > XY, BC > YZ, and CA > ZX . By what theorem or postulate can you conclude that nABC > nXYZ?

Get Ready! To prepare for Lesson 5-7, do Exercises 53–55.

Use the fi gure at the right.

53. What is m/P?

54. What is m/D?

55. Is it possible for AW to equal OG?

See Lesson 5-5.

See Lesson 4-2.

See Lesson 3-5.A O

D G

P W

35 30

105 125

Standardized Test Prep

46. Th e fi gure shows the walkways connecting four dormitories on a college campus. What is the greatest possible whole-number length, in yards, for the walkway between South dorm and East dorm?

47. What is the length of a segment with endpoints A(213, 216) and B(29, 213)?

48. How many sides does a convex quadrilateral have?

49. /1 and /2 are corresponding angles formed by two parallel lines and a transversal. If m/1 5 33x 1 2 and m/2 5 68, what is the value of x?

SAT/ACT

West

South

42 yd

31 yd

57 yd

North

East