extends mafs.912.g-co.3.10 prove theorems inequalities in ... · objective to apply inequalities in...

57 Inequalities in TwoTriangles

Objective To apply inequalities in two triangles

Mathematics Florida StandardsExtends MAFS.912.G-CO.3.10 Prove theorems

about triangles

MP1,MP3

Getting Ready! X C A

Try to make theproblem simplerby finding a way tofind the distance x

without measuringdirectly.

Think of a clock or watch that has an hour hand and

a minute hand. As minutes pass, the distance betweenthe tip of the hour hand and the tip of the minutehand changes. This distance is x in the figure at theright. What is the order of the times below fromleast to greatest length of x? How do you know?

1:00, 3:00, 5:00, 8:30, 1:30, 12:20

MATHEMATICAL

PRACTICES jjj Solve It, the hands of the clock and the segment labeled x form a triangle. As thetime changes, the shape of the triangle changes, but the lengths of two of its sides do

not change.

Essential Understanding in triangles that have two pairs of congruent sides,there is a relationship between the included angles and the third pair of sides.

When you close a door, the angle between the

door and the frame (at the hinge) gets smaller.The relationship between the measure of the

hinge angle and the length of the opposite side

is the basis for the SAS Inequality Theorem, also

known as the Hinge Theorem.

Theorem 5-13 The Hinge Theorem (SAS Inequality Theorem)

Theorem

If two sides of one triangle

are congruent to two sides

of another triangle, and

the included angles are not

congruent, then the longer

third side is opposite the

larger included angle.

m^A > m^X

Then...

BOYZ

You will prove Theorem 5-13 in Exercise 25.

332 Chapter 5 Relationships Within Triangles

How do you applythe Hinge Theorem?After you identify theangles included betweenthe pairs of congruentsides, locate the sidesopposite those angles.

For which side

lengths are the sameat Time 1 and Time 2?

The lengths of the chain

and AB do not change.So, AS and 8C are the

same at Time 1 and

Time 2.

Using the Hinge Theorem

Multiple Choice Which of the following

statements must be true?

Ca:> AS< YU

ct:) sk> yu

cc::>sK<Yu

CE) AK= YU

SA = YO and AK = OU, so the triangles have two pairs of congruent sides.

The included angles, LA and LO, are not congruent. Since mLA > mLO,

SK > YU by the Hinge Theorem. Hie correct answer is B.

Got It? 1. a. What inequality relates LN and OQin tlie figure at the right?

b. Reasoning In AABC, AB = 3,BC = 4, and CA = 6. In APQR,

PQ = S,QR^ 5, and RP = 6.

How can you use indirect reasoning

to explain why mLP > mLAl

Time 1 Time 2

Applying the Hinge Theorem

Swing Ride The diagram below shows the position of a

swing at two different times. As the speed of the swing

ride increases, the angle between the chain and AB

increases. Is the rider farther from point A at

Time 1 or Time 2? Explain how the Hinge

Theorem justifies your answer.

The rider is farther from point A at Time 2.

The lengths of AB and BC stay the same

throughout the ride. Since the angle

formed at Time 2 {L2) is greater thanthe angle formed at Time 1 [LI],you can use the Hinge Theorem to

conclude that AC at Time 2 is longer

tlian AC at Time 1.

Lesson 5-7 Inequalities in Two Triangles 333

Gofit? 2. The diagram below shows a pair of scissors in two different positions. Inwhich position is the distance between the tips of the two blades greater?Use the Hinge Theorem to justily your answer.

The Converse of the Hinge Theorem is also true. The proof of the converse is anindirect proof.

Theorem 5-14 Converse of the Hinge Theorem {SSS Inequality)

Theorem if... Then ...

If two sides of one triangle

are congruent to two sides

of another triangle, and

the third sides are not

congruent, then the larger

included angle is opposite

the longer third side.

SO yz mZ^A > mLX

Proof Indirect Proof of the Converse of the Hinge Theorem (SSS Inequality)

Given: AB = ̂,AC = XZ,BO yz

Prove: mAA > mAX

Step 1 Assume temporarily that mAA > mAX.

This means either mAA < mAJC or

mAA = mAX.

Step 2 If mAA < mAX, then EC < FZ by the Hinge Theorem. This contradicts

the given information that EC > YZ. Therefore, the assumption thatmAA < mAX must be false.

If mAA = mAX, then AABC = AXYZ by SAS. If the two triangles arecongruent, then EC — YZ because corresponding parts of congruent

triangles are congruent. This contradicts the given information that EC > YZ.

Therefore, the assumption that mAA = mAX must be false.

Step 3 The temporary assumption that mAA > mAAX is false. Therefore,

mAA > mAJC.


I-*" ai rt n

How do you putupper and lowerlimits on the value

of X?

Use the largest possiblevalue oi mz.TUS as the

upper limit for Sx - 20and the smallest possiblevalue ofm/LTUSas the

lower limit for 5x-20.

Using the Converse of the Hinge Theorem

Algebra What is the range of possible values for x?

Step 1 Find an upper limit for the value of x. UT = UR andUS = US, so A TUS and ARUS have two pairs of

congruent sides. RS > TS, so you can use the Converseof the Hinge Theorem to write an inequality.

m/LRUS > ml. TUS Converse of the Hinge Theorem

60 > Sx — 20 Substitute.

80 > 5x Add 20 to each side.

16 > jc Divide each side by 5.

Find a lower limit for the value of x.Step 2

10 '

mATUS >0 The measure of an angle of a triangle is greater than 0.

Sx — 20 > 0 Substitute.

5x > 20 Add 20 to each side.

x> 4 Divide each side by 5.

Rewrite 16 > x and x > 4 as 4 < x < 16.

Got It? 3. What is the range of possible values Tfor X in the figure at the right?

(3x + 18)

Proof

Th'rI.

How do you knowmlBAE > mLBEA7

Use the Transitive

Property of Inequalityon the inequalities InStatements 4 and 6.

Problem 4 Proving Relationships in Triangles

Given: BA = DE, BE > DA

Prove: mABAE > mABEA

Statement Reasons

1) BA = DE 1) Given

2) AE = AE 2) Reflexive Property of Equality

3) BE>DA 3) Given

4) mABAE > mADEA 4) Converse of the Hinge Theorem

5) mADEA = mADEB + mABEA 5) Angle Addition Postulate

6) mADEA > mABEA 6) Comparison Property of Inequality

7) mABAE > mABEA 7) Transitive Property of Inequality

Got It? 4. Given: = 80, O is the midpoint of LAf

Prove: LM>MN

c PowerGeometry.com \ Lesson 5-7 Inequalities in Two Triangles 335

&Lesson Check

Do you know HOW?

Write an inequality relating the givenside lengths or angle measures.

1.FDandjBC B £ ], p

2. mzit/Sr and mAVST

D

14

12

_ . MATHEMATICALDo you UNDERSTAND? PRACTICES

3. Vocabuiary Explain why Hmge Theorem is an

appropriate name for Theorem 5-13.

4. Error Analysis From the

figure at the right, your

friend concludes that

m/LBAD > m/LBCD. How

would you correct your

friend's mistake?

5. Compare and Contrast How are the Hinge Theorem

and the SAS Congruence Postulate similar?

MATHEMATICAL

PRACTICESPractice and Problem-Solving Exercises

Practice Write an inequality relating the given side lengths. If there is not enoughinformation to reach a conclusion, write no conclusion.

6. AB and AD

8. LM and KL

7. PR and RT

^ See Problem 1.

9. yz and UV

10. The diagram below shows a robotic arm in two different positions. In which ^ See Problem 2.position is the tip of the robotic arm closer to the base? Use the Hinge Theorem

to justify your answer.


Algebra Find the range of possible values for each variable. ^ See Problem 3.

11.{X - 6)

13.

36

(2X-12)

4X-10

^ Apply

15. Developing Proof Complete the following proof.

Given; Cis the midpoint of BD,

m/LEAC = rri/LAEC,

mZ-BCA > m/LDCE

Prove: AB>ED

Statements Reasons

1) m/.EAC = mZ-AEC 1) Given

2) AC=EC 2) a. J_

3) C is the midpoint of 5D. 3) b. J_

4) IC = ̂ 4) c. J_

5) d- 5) = segments have = length.

6) m/LBCA > m/-DCE 6) e. J_

7) AB>ED 7) f. J_

Copy and complete with > or <. Explain your reasoning.

16. PTBQfi Q

17. mAQTRmm/.RTS

18. PTMRS

^ See Problem 4.

19. a. Error Analysis Your classmate draws the figure at the right. Explainwhy the figure cannot have the labeled dimensions,

b. Open-Ended Describe a way you could change the dimensions tomake the figure possible.

C PowerGeometry.cbm Lesson 5-7 Inequalities in Two Triangles 337

start

20. Think About a Plan Ship A and Ship B leave from the

same point in the ocean. Ship A travels 150 mi due west,

turns 65° toward north, and then travels another 100 mi.

Ship B travels 150 mi due east, turns 70° toward south, V

and then travels another 100 mi. Which ship is farther

from the starting point? Explain.

• How can you use the given angle measures?

• How does the Hinge Theorem help you to solve

this problem?

21. Which of the following lists the segment lengths in order

from least to greatest?

CS> CD, AB, DE, BC, EE

CT) EE, DE, AB, BC, CD

CO BC, DE, EE, AB. CD

cO EE, BC, DE, AB, CD

22. Reasoning The legs of a right isosceles triangle are congruent to the legs of anisosceles triangle with an 80° vertex angle. Which triangle has a greater perimeter?How do you know?

23. Use the figure at the right.Proof Qjygp. is isosceles with vertex AB,

AABE = ACBD,

mAEBD > mAABE

Prove: ED>AE

j^Chaiienqe 24. Coordinate Geometry AABC has vertices A(0, 7), B(-l, —2), C(2, —1), and0(0, 0). Show that mAAOB > mAAOC.

25. Use the plan below to complete a proof of the Hinge Theorem: If two sides of oneProof triangle are congruent to two sides of another triangle and the included angles are

not congruent, then the longer third side is opposite the larger included angle.

Given: AB = XY, BC = YZ, mAB > mAY

Prove: AOXZ.

Plan for proof:

• Copy AABC. Locate point D outside AABC so

that mACBD — mAZYX and BD = YX. Show that

ADBC = tJCYZ.

• Locate point Fon AC, so that BE bisects AABD.

• Show that AABE = ADBE and that AE = DE.

• Show that AC = FC +DF.

• Use the Triangle Inequality Theorem to write an

inequality that relates DC to the lengths of the other

sides of AECD.

• Relate DC and XZ.

H

s

ShipB

X


,l©lApply What You've Learned

Look at the trail map from page 283, shown again below. In the Apply What

You've Learned sections in Lessons 5-1 and 5-6, you worked with the first

two segments of the hike from the campground to the waterfall. Now you will

consider the last segment of the hike.

MATHEMATICAL

PRACTICES

MP2

2.5 km

Campground

C

4.2 km

3.5 km 4.8 km

3.1 km

5.1 km

■""..^WaterfaliW

3.2 km

a. Write an inequality that shows an upper bound for the length of LW.Explain your reasoning.

b. Write an inequality that shows a lower bound for the length of LW.Explain your reasoning.

c. Use your answers from parts (a) and (b) to write a compound inequalitythat shows the range of possible lengths of LW.

c PowerGeometry.com Lesson 5-7 Inequalities in Two Triangles 339

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