Vocabulary - Mr. Yildiz Mathematics...

284 Chapter 4 Triangle Congruence

For a complete list of the postulates and theorems in this chapter, see p. S82.

acute triangle . . . . . . . . . . . . . . 216

auxiliary line . . . . . . . . . . . . . . . 223

base . . . . . . . . . . . . . . . . . . . . . . . 273

base angle . . . . . . . . . . . . . . . . . . 273

congruent polygons . . . . . . . . . 231

coordinate proof . . . . . . . . . . . . 267

corollary . . . . . . . . . . . . . . . . . . . 224

corresponding angles . . . . . . . 231

corresponding sides . . . . . . . . . 231

CPCTC . . . . . . . . . . . . . . . . . . . . . 260

equiangular triangle . . . . . . . . 216

equilateral triangle . . . . . . . . . 217

exterior . . . . . . . . . . . . . . . . . . . . 225

exterior angle . . . . . . . . . . . . . . 225

included angle . . . . . . . . . . . . . . 242

included side . . . . . . . . . . . . . . . 252

interior . . . . . . . . . . . . . . . . . . . . 225

interior angle . . . . . . . . . . . . . . . 225

isosceles triangle . . . . . . . . . . . 217

legs of an isosceles triangle . . 273

obtuse triangle . . . . . . . . . . . . . 216

remote interior angle . . . . . . . 225

right triangle . . . . . . . . . . . . . . . 216

scalene triangle . . . . . . . . . . . . . 217

triangle rigidity . . . . . . . . . . . . . 242

vertex angle . . . . . . . . . . . . . . . . 273

Complete the sentences below with vocabulary words from the list above.

1. A(n) −−−− ? is a triangle with at least two congruent sides.

2. A name given to matching angles of congruent triangles is −−−− ? .

3. A(n) −−−− ? is the common side of two consecutive angles in a polygon.

Classify each triangle by its angle measures and side lengths.

4. 5.

■ Classify the triangle by its angle measures and side lengths.

isosceles right triangle

4-1 Classifying Triangles (pp. 216–221)

EXERCISESE X A M P L E

Find m∠N.

6.

7. In�LMN, m∠L = 8x °, m∠M = (2x + 1) °, and m∠N = (6x - 1) °.

■ Find m∠S. 12x = 3x + 42 + 6x

12x = 9x + 42

3x = 42

x = 14

m∠S = 6 (14) = 84°

4-2 Angle Relationships in Triangles (pp. 223–230)


Vocabulary

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Study Guide: Review 285

Given: �PQR � �XYZ. Identify the congruent corresponding parts.

8. −−

PR � −−−− ? 9. ∠Y � −−− ?

Given: �ABC � �CDA Find each value.

10. x

11. CD

■ Given: �DEF � �JKL. Identify all pairs of congruent corresponding parts. Then find the value of x.

The congruent pairs follow: ∠D � ∠J, ∠E � ∠K, ∠F � ∠L,

−− DE �

−− JK ,

−− EF �

−− KL , and

−− DF �

−− JL .

Since m∠E = m∠K, 90 = 8x - 22. After 22 is added to both sides, 112 = 8x. So x = 14.

4-3 Congruent Triangles (pp. 231–237)


12. Given: −−

AB � −−

DE , −−

DB � −−

AE Prove: �ADB � �DAE

13. Given: −−

GJ bisects −−

FH , and

−− FH bisects

−− GJ .

Prove: �FGK � �HJK

14. Show that �ABC � �XYZ when x = -6.

15. Show that �LMN � �PQR when y = 25.

■ Given: −−

RS � −−

UT , and −−

VS � −−

VT . V is the midpoint of

−− RU .

Prove: �RSV � �UTV

Proof:

Statements Reasons

1. −−

RS � −−

UT

2. −−

VS � −−

VT

3. V is the mdpt. of −−

RU .

4. −−

RV � −−

UV

5. �RSV � �UTV

1. Given

2. Given

3. Given

4. Def. of mdpt.

5. SSS Steps 1, 2, 4

■ Show that �ADB � �CDB when s = 5.

AB = s 2 - 4s AD = 14 - 2s

= 5 2 - 4 (5 ) = 14 - 2 (5 )

= 5 = 4

−−

BD � −−

BD by the Reflexive Property. −−

AD � −−

CD and

−− AB �

−− CB . So �ADB � �CDB by SSS.

4-4 Triangle Congruence: SSS and SAS (pp. 242–249)

EXERCISESE X A M P L E S



16. Given: C is the midpoint of

−− AG .

−−

HA ‖ −−

GB Prove: �HAC � �BGC

17. Given: −−−

WX ⊥ −−

XZ , −−

YZ ⊥ −−

ZX , −−−

WZ � −−

YX Prove: �WZX � �YXZ

18. Given: ∠S and ∠V are right angles.RT = UW.m∠T = m∠W

Prove: �RST � �UVW

■ Given: B is the midpoint of −−

AE . ∠A � ∠E,∠ABC � ∠EBD

Prove: �ABC � �EBD

Proof:

Statements Reasons

1. ∠A � ∠E

2. ∠ABC � ∠EBD

3. B is the mdpt. of −−

AE .

4. −−

AB � −−

EB

5. �ABC � �EBD

1. Given

2. Given

3. Given

4. Def. of mdpt.

5. ASA Steps 1, 4, 2

4-5 Triangle Congruence: ASA, AAS, and HL (pp. 252–259)


19. Given: M is the midpoint of

−− BD .

−−

BC � −−

DC Prove: ∠1 � ∠2

20. Given: −−

PQ � −−

RQ , −−

PS � −−

RS Prove:

−− QS bisects ∠PQR.

21. Given: H is the midpoint of −−

GJ . L is the midpoint of

−−− MK .

−−−

GM � −−

KJ , −−

GJ � −−−

KM ,∠G � ∠K

Prove: ∠GMH � ∠KJL

■ Given: −−JL and

−−HK bisect each other.

Prove: ∠JHG � ∠LKG

Proof:

Statements Reasons

1. −−

JL and −−

HK bisect each other.

2. −−

JG � −−

LG , and

−−− HG �

−− KG .

3. ∠JGH � ∠LGK

4. �JHG � �LKG

5. ∠JHG � ∠LKG

1. Given

2. Def. of bisect

3. Vert. � Thm.

4. SAS Steps 2, 3

5. CPCTC

4-6 Triangle Congruence: CPCTC (pp. 260–265)




Position each figure in the coordinate plane and give the coordinates of each vertex.

22. a right triangle with leg lengths r and s

23. a rectangle with length 2p and width p

24. a square with side length 8m

For exercises 25 and 26 assign coordinates to each vertex and write a coordinate proof.

25. Given: In rectangle ABCD, E is the midpoint of −−AB , F is the midpoint of

−−BC , G is the

midpoint of −−CD , and H is the midpoint

of −−AD .

Prove:−−EF �

−−−GH

26. Given: �PQR has a right ∠Q . M is the midpoint of

−− PR .

Prove: MP = MQ = MR

27. Show that a triangle with vertices at (3, 5) , (3, 2) , and (2, 5) is a right triangle.

■ Given: ∠B is a right angle in isosceles right�ABC. E is the midpoint of

−− AB .

D is the midpoint of −−CB .

−−AB �

−−CB

Prove:−−CE �

−−AD

Proof: Use the coordinates A(0, 2a) , B(0, 0) ,and C (2a, 0) . Draw

−− AD and

−− CE .

By the Midpoint Formula,

E = ( 0 + 0 _ 2

, 2a + 0 _ 2

) = (0, a) and

D = ( 0 + 2a _ 2

, 0 + 0 _ 2

) = (a, 0)

By the Distance Formula,

CE = √ (2a - 0) 2 + (0 - a) 2

= √ 4a 2 + a 2 = a √ 5

AD = √ (a - 0) 2 + (0 - 2a) 2

= √ a 2 + 4a 2 = a √ 5

Thus −−

CE � −−

AD by the definition of congruence.

4-7 Introduction to Coordinate Proof (pp. 267–272)


Find each value.

28. x

29. RS

30. Given: �ACD is isosceles with ∠D as the vertex angle. B is the midpoint of

−− AC .

AB = x + 5, BC = 2x - 3, and CD = 2x + 6. Find the perimeter of �ACD.

■ Find the value of x.

m∠D + m∠E + m∠F = 180° by the Triangle Sum Theorem. m∠E = m∠F by the Isosceles Triangle Theorem.

m∠D + 2 m∠E = 180° Substitution

Substitute the given values.

Simplify.

Divide both sides by 6.

42 + 2 (3x) = 180

6x = 138

x = 23

4-8 Isosceles and Equilateral Triangles (pp. 273–279)




1. Classify �ACD by its angle measures.

Classify each triangle by its side lengths.

2. �ACD 3. �ABC 4. �ABD

5. While surveying the triangular plot of land shown, a surveyor finds that m∠S = 43°. The measure of ∠RTP is twice that of ∠RTS. What is m∠R?

Given: �XYZ � �JKL Identify the congruent corresponding parts.

6. −−

JL � −−−− ? 7. ∠Y � −−−− ? 8. ∠L � −−−− ? 9. −−

YZ � −−−− ?

10. Given: T is the midpoint of −−

PR and −−

SQ . Prove: �PTS � �RTQ

11. The figure represents a walkway with triangular supports. Given that

−− GJ bisects

∠HGK and ∠H � ∠K, use AAS to prove �HGJ � �KGJ

12. Given: −−

AB � −−

DC , 13. Given: −−

PQ ‖ −−

SR ,

−− AB ⊥

−− AC , ∠S � ∠Q

−−

DC ⊥ −−

DB Prove: −−

PS ‖ −−

QR Prove: �ABC � �DCB

14. Position a right triangle with legs 3 m and 4 m long in the coordinate plane. Give the coordinates of each vertex.

15. Assign coordinates to each vertex and write a coordinate proof.

Given: Square ABCDProve:

−− AC �

−− BD

Find each value.

16. y 17. m∠S

18. Given: Isosceles �ABC has coordinates A (2a, 0) , B (0, 2b) , and C (-2a, 0) . D is the midpoint of

−− AC , and E is the midpoint of

−− AB .

Prove: �AED is isosceles.


FOCUS ON ACTThe ACT Mathematics Test is one of four tests in the ACT. You are given 60 minutes to answer 60 multiple-choice questions. The questions cover material typically taught through the end of eleventh grade. You will need to know basic formulas but nothing too difficult.

You may want to time yourself as you take this practice test. It should take you about 5 minutes to complete.

There is no penalty for guessing on the ACT. If you are unsure of the correct answer, eliminate as many answer choices as possible and make your best guess. Make sure you have entered an answer for every question before time runs out.

College Entrance Exam Practice 289

1. For the figure below, which of the following must be true?

I. m∠EFG > m∠DEF

II. m∠EDF = m∠EFD

III. m∠DEF + m∠EDF > m∠EFG

(A) I only

(B) II only

(C) I and II only

(D) II and III only

(E) I, II, and III

2. In the figure below, �ABD � �CDB, m∠A = (2x + 14) °, m∠C = (3x - 15) °, and m∠DBA = 49°. What is the measure of ∠BDA?

(F) 29°

(G) 49°

(H) 59°

(J) 72°

(K) 101°

3. Which of the following best describes a triangle with vertices having coordinates (-1, 0) , (0, 3) , and (1, -4) ?

(A) Equilateral

(B) Isosceles

(C) Right

(D) Scalene

(E) Equiangular

4. In the figure below, what is the value of y?

(F) 49

(G) 87

(H) 93

(J) 131

(K) 136

5. In �RST, RS = 2x + 10, ST = 3x - 2, and RT = 1 __

2 x + 28. If �RST is equiangular, what

is the value of x?

(A) 2

(B) 5 1 _ 3

(C) 6

(D) 12

(E) 34


Test Tackler 291

Read each test item and answer the questions that follow.

Scoring Rubric:4 points: The student demonstrates a thorough understanding of the concept, correctly answers the question, and provides a complete explanation.

3 points: The student correctly answers the question but does not show all work or does not provide an explanation.

2 points: The student makes minor errors resulting in an incorrect solution but shows and explains an understanding of the concept.

1 point: The student gives a response showing no work or explanation.

0 points: The student gives no response.

Item AWhat theorem(s) can you use, other than the HL Theorem, to prove that �MNP � �XYZ ? Explain your reasoning.

1. What should a full-credit response to this test item include?

2. A student wrote this response:

What score should this response receive? Why?

3. Write a list of the ways to prove triangles congruent. Is the Pythagorean Theorem on your list?

4. Add to the response so that it receives a score of 4-points.

Item BCan an equilateral triangle be an obtuse triangle? Explain your answer. Include a sketch to support your reasoning.

5. What should a full-credit response to this test item include?

6. A student wrote this response:

Why will this response not receive a score of 4 points?

7. Correct the response so that it receives full credit.

Item CAn isosceles right triangle has two sides, each with length y + 4.

Describe how you would find the length of the hypotenuse. Provide a sketch in your explanation.

8. A student began trying to find the length of the hypotenuse by writing the following:

Is the student on his way to receiving a 4-point response? Explain.

9. Describe a different method the student could use for this response.

To receive full credit, make sure all parts of the problem are answered. Be sure to provide a complete explanation for your reasoning.



KEYWORD: MG7 TestPrep

CUMULATIVE ASSESSMENT, CHAPTERS 1–4

Multiple Choice

Use the diagram for Items 1 and 2.

C

D

AE

B

1. Which of these congruence statements can be proved from the information given in the figure?

�AEB � �CED �ABD � �BCA

�BAC � �DAC �DEC � �DEA

2. What other information is needed to prove that �CEB � �AED by the HL Congruence Theorem?

−AD �

−AB

−CB �

−AD

−BE �

−AE

−DE �

−CE

3. Which biconditional statement is true?

Tomorrow is Monday if and only if today is not Saturday.

Next month is January if and only if this month is December.

Today is a weekend day if and only if yesterday was Friday.

This month had 31 days if and only if last month had 30 days.

4. What must be true if �⎯�PQ intersects

�⎯�ST at more

than one point?

P, Q, S, and T are collinear.

P, Q, S, and T are noncoplanar.⎯�PQ and

⎯�ST are opposite rays.

�⎯�PQ and

�⎯�ST are perpendicular.

5. �ABC � �DEF, EF = x 2 - 7, and BC = 4x - 2. Find the values of x.

-1 and 5 1 and 5

-1 and 6 2 and 3

6. Which conditional statement has the same truth value as its inverse?

If n < 0, then n 2 > 0.

If a triangle has three congruent sides, then it is an isosceles triangle.

If an angle measures less than 90°, then it is an acute angle.

If n is a negative integer, then n < 0.

7. On a map, an island has coordinates (3, 5) , and a reef has coordinates (6, 8) . If each map unit represents 1 mile, what is the distance between the island and the reef to the nearest tenth of a mile?

4.2 miles 9.0 miles

6.0 miles 15.8 miles

8. A line has an x-intercept of -8 and a y-intercept of 3. What is the equation of the line?

y = -8x + 3 y = 8 _ 3

x - 8

y = 3 _ 8 x + 3 y = 3x - 8

9. �

⎯�

JK passes through points J (1, 3) and K (-3, 11) . Which of these lines is perpendicular to

�

⎯� JK ?

y = - 1 _ 2 x + 1 _

3 y = -2x - 1 _

5

y = 1 _ 2 x + 6 y = 2x - 4

10. If PQ = 2 (RS) + 4 and RS = TU + 1, which equation is true by the Substitution Property of Equality?

PQ = TU + 5

PQ = TU + 6

PQ = 2 (TU) + 5

PQ = 2 (TU) + 6

11. Which of the following is NOT valid for proving that triangles are congruent?

AAA SAS

ASA HL

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Cumulative Assessment, Chapters 1–4 293

Use this diagram for Items 12 and 13.

C

DA

EB

100˚

12. What is the measure of ∠ACD?

40° 100°

80° 140°

13. What type of triangle is �ABC?

Isosceles acute

Equilateral acute

Isosceles obtuse

Scalene acute

Gridded Response 14. �CDE � �JKL. m∠E = (3x + 4) °, and

m∠L = (6x - 5) °. What is the value of x?

15. Lucy, Eduardo, Carmen, and Frank live on the same street. Eduardo’s house is halfway between Lucy’s house and Frank’s house. Lucy’s house is halfway between Carmen’s house and Frank’s house. If the distance between Eduardo’s house and Lucy’s house is 150 ft, what is the distance in feet between Carmen’s house and Eduardo’s house?

16. �JKL � �XYZ, and JK = 10 - 2n. XY = 2, and YZ = n 2 . Find KL.

17. An angle is its own supplement. What is its measure?

18. The area of a circle is 154 square inches. What is its circumference to the nearest inch?

19. The measure of ∠P is 3 1 __ 2 times the measure of ∠Q.

If ∠P and ∠Q are complementary, what is m∠P in degrees?

Short Response 20. Given � ‖ m with transversal n, explain why ∠2

and ∠3 are complementary.

n

m

�

12

3

21. ∠G and ∠H are supplementary angles. m∠G = (2x + 12) °, and m∠H = x°.

a. Write an equation that can be used to determine the value of x. Solve the equation and justify each step.

b. Explain why ∠H has a complement but ∠G does not.

22. A manager conjectures that for every 1000 parts a factory produces, 60 are defective.

a. If the factory produces 1500 parts in one day, how many of them can be expected to be defective based on the manager’s conjecture? Explain how you found your answer.

b. Use the data in the table below to show that the manager’s conjecture is false.

Day 1 2 3 4 5

Parts 1000 2000 500 1500 2500

DefectiveParts

60 150 30 90 150

23. −

BD is the perpendicular bisector of −

AC .

a. What are the conclusions you can make from this statement?

b. Suppose −−

BD intersects −−

AC at D. Explain why −−

BD is the shortest path from B to

−− AC .

Extended Response 24. �ABC and �DEF are isosceles triangles.

−− BC �

−− EF ,

and −−

AC � −−

DF . m∠C = 42.5°, and m∠E = 95°.

a. What is m∠D? Explain how you determined your answer.

b. Show that �ABC and �DEF are congruent.

c. Given that EF = 2x + 7 and AB = 3x + 2, find the value for x. Explain how you determined your answer.

Take some time to learn the directions for filling in a grid. Check and recheck to make sure you are filling in the grid properly. You will only get credit if the ovals below the boxes are filled in correctly.To check your answer, solve the problem using a different method from the one you originally used. If you made a mistake the first time, you are unlikely to make the same mistake when you solve a different way.

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366 Chapter 5 Properties and Attributes of Triangles

For a complete list of the postulates and theorems in this chapter, see p. S82.

altitude of a triangle . . . . . . . . 316

centroid of a triangle . . . . . . . . 314

circumcenter of a triangle . . . 307

circumscribed . . . . . . . . . . . . . . 308

concurrent . . . . . . . . . . . . . . . . . 307

equidistant . . . . . . . . . . . . . . . . . 300

incenter of a triangle . . . . . . . . 309

indirect proof . . . . . . . . . . . . . . . 332

inscribed . . . . . . . . . . . . . . . . . . . 309

locus . . . . . . . . . . . . . . . . . . . . . . . 300

median of a triangle . . . . . . . . 314

midsegment of a triangle . . . . 322

orthocenter of a triangle . . . . 316

point of concurrency . . . . . . . . 307

Pythagorean triple . . . . . . . . . . 349

Complete the sentences below with vocabulary words from the list above.

1. A point that is the same distance from two or more objects is −−−− ? from the objects.

2. A −−−− ? is a segment that joins the midpoints of two sides of the triangle.

3. The point of concurrency of the angle bisectors of a triangle is the −−−− ? .

4. A −−−− ? is a set of points that satisfies a given condition.

Find each measure.

5. BD 6. YZ

7. HT 8. m∠MNP

Write an equation in point-slope form for the perpendicular bisector of the segment with the given endpoints.

9. A (-4, 5) , B (6, -5) 10. X (3, 2) , Y (5, 10)

Tell whether the given information allows you to conclude that P is on the bisector of ∠ABC.

11. 12.

Find each measure.

■ JL

Because −−

JM � −−−

MK and −−−

ML ⊥ −−

JK , −−−

ML is the perpendicular bisector of

−− JK .

JL = KL ⊥ Bisector Thm.

Substitute 7.9 for KL.JL = 7.9

■ m∠PQS, given that m∠PQR = 68°

Since SP = SR, −−

SP ⊥ −−

QP , and

−− SR ⊥

−− QR , �� QS bisects

∠PQR by the Converse of the Angle Bisector Theorem.

m∠PQS = 1 _ 2

m∠PQR Def. of ∠ bisector

Substitute 68° for m∠PQR.

m∠PQS = 1 _ 2

(68°) = 34°

5-1 Perpendicular and Angle Bisectors (pp. 300–306)


Vocabulary



−− PX ,

−− PY , and

−− PZ are the

perpendicular bisectors of �GHJ. Find each length.

13. GY 14. GP

15. GJ 16. PH

−− UA and

−− VA are angle

bisectors of �UVW. Find each measure.

17. the distance from A to

−− UV

18. m∠WVA

Find the circumcenter of a triangle with the given vertices.

19. M (0, 6) , N (8, 0) , O (0, 0)

20. O (0, 0) , R (0, -7) , S (-12, 0)

■ −−

DG , −−

EG , and −−

FG are the perpendicular bisectors of �ABC. Find AG.

G is the circumcenter of �ABC. By the Circumcenter Theorem, G is equidistant from the vertices of �ABC.

AG = CG Circumcenter Thm.

Substitute 5.1 for CG.AG = 5.1

■ −−

QS and −−

RS are angle bisectors of �PQR. Find the distance from S to

−− PR .

S is the incenter of �PQR. By the Incenter Theorem, S is equidistant from the sides of �PQR. The distance from S to

−− PQ is 17, so the

distance from S to −−

PR is also 17.

5-2 Bisectors of Triangles (pp. 307–313)


In �DEF, DB = 24.6, and EZ = 11.6. Find each length.

21. DZ 22. ZB

23. ZC 24. EC

Find the orthocenter of a triangle with the given vertices.

25. J (-6, 7) , K (-6, 0) , L (-11, 0)

26. A (1, 2) , B (6, 2) , C (1, -8)

27. R (2, 3) , S (7, 8) , T (8, 3)

28. X (-3, 2) , Y (5, 2) , Z (3, -4)

29. The coordinates of a triangular piece of a mobile are (0, 4) , (3, 8) , and (6, 0) . The piece will hang from a chain so that it is balanced. At what coordinates should the chain be attached?

■ In �JKL, JP = 42. Find JQ.

JQ = 2_3

JP Centroid Thm.

Substitute 42 for JP.

Multiply.

JQ = 2_3

(42)

JQ = 28

■ Find the orthocenter of �RST with vertices R (-5, 3) , S (-2, 5) , and T(-2, 0) .

Since −−ST is vertical, the

equation of the line containing the altitude from R to

−− ST is y = 3.

slope of −−

RT = 3 - 0 _ -5 - (-2)

= -1

The slope of the altitude to −−

RT is 1. This line must pass through S (-2, 5) .

y - y 1 = m (x - x 1 ) Point-slope form

Substitution y - 5 = 1 (x + 2)

Solve the system

y = 3

y = x + 7

to find that the

coordinates of the orthocenter are (-4, 3) .

5-3 Medians and Altitudes of Triangles (pp. 314–320)



368 Chapter 5 Properties and Attributes of Triangles

Find each measure.

30. BC 31. XZ

32. XC 33. m∠BCZ

34. m∠BAX 35. m∠YXZ

36. The vertices of �GHJ are G (-4, -7) , H (2, 5) , and J (10, -3) . V is the midpoint of

−−− GH , and

W is the midpoint of −−

HJ . Show that −−−

VW ‖ −−

GJ and VW = 1 __

2 GJ.

Find each measure.

■ NQ

By the � Midsegment

Thm., NQ = 1 _ 2

KL = 45.7.

■ m∠NQM

−−

NP ‖ −−−

ML � Midsegment Thm.Alt. Int. � Thm.Substitution

m∠NQM = m∠PNQ m∠NQM = 37°

5-4 The Triangle Midsegment Theorem (pp. 322–327)


37. Write the sides of �ABC in order from shortest to longest.

38. Write the angles of �FGH in order from smallest to largest.

39. The lengths of two sides of a triangle are 13.5 centimeters and 4.5 centimeters. Find the range of possible lengths for the third side.

Tell whether a triangle can have sides with the given lengths. Explain.

40. 6.2, 8.1, 14.2 41. z, z, 3z, when z = 5

42. Write an indirect proof that a triangle cannot have two obtuse angles.

■ Write the angles of �RSTin order from smallest to largest.

The smallest angle is opposite the shortest side. In order, the angles are ∠S, ∠R, and ∠T.

■ The lengths of two sides of a triangle are 15 inches and 12 inches. Find the range of possible lengths for the third side.

Let s be the length of the third side.

s + 15 > 12 s + 12 > 15 15 + 12 > s s > -3 s > 3 27 > s

By the Triangle Inequality Theorem, 3 in. < s < 27 in.

5-5 Indirect Proof and Inequalities in One Triangle (pp. 332–339)


Compare the given measures.

43. PS and RS 44. m∠BCA and m∠DCA

Find the range of values for n.

45. 46.

Compare the given measures.

■ KL and ST

KJ = RS, JL = RT, and m∠J > m∠R. By the Hinge Theorem, KL > ST.

■ m∠ZXY and m∠XZW

XY = WZ, XZ = XZ, and YZ < XW. By the Converse of the Hinge Theorem, m∠ZXY < m∠XZW.

5-6 Inequalities in Two Triangles (pp. 340–345)



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Vocabulary - Mr. Yildiz Mathematics...

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