46 Congruence in RightTriangles

Mathematics Florida StandardsMAFS.912.G-SRT.2.5 Use congruence... aiteria tosolve problems and prove relationships in geometricfigures.

MP1,MP3

Objective To prove right triangles congruent using the Hypotenuse-Leg Theorem

Getting Ready!

What does the

large triangle tellyou about angles inthe figure?

MATHEMATICAL

PRACTICES

In the diagram below, two sides and a nonincluded angle of one triangle are congruent

to two sides and the nonincluded angle of another triangle.

Q

Lesson

Vocabularyhypotenuselegs of a righttriangle

J

Notice that the triangles are not congruent. So, you can conclude that Side-Side-Angle

is not a valid method for proving two triangles congruent. This method, however, works

in the special case of right triangles, where the right angles are the nonincluded angles.

In a right triangle, the side opposite the

right angle is called the hypotenuse. It is

the longest side in the triangle. The other

two sides are called legs.

Tlie right anglealways "points" tothe hypotenuse. J

Hypotenuse

Essential Understanding You can prove that two triangles are congruentwithout having to show that all corresponding parts are congruent. In this lesson,

you will prove right triangles congruent by using one pair of right angles, a pair of

hypotenuses, and a pair of legs.

258 Chapter 4 Congruent Triangles

Theorem 4-6 Hypotenuse-Leg (HI) Theorem

Theorem

If the hypotenuse

and a leg of one right

triangle are congruent

to the hypotenuse

and a leg of another

right triangle, then

the triangles are

congruent.

If...

APQR and AXYZ are right A,

^ = )a,and^ = W

Then ...

APQR = AXYZ

To prove the HL Theorem you will need to draw auxiliary lines to make a third triangle.

Pr^f Proof of Theorem 4-6: Hypotenuse-Leg Theorem

Given: APQR and AXYZ are right triangles, with right angles

Q and K ̂ ̂ and PQ = )^,

Prove: APQR = AXYZ

Proof: On AXYZ, draw ZY.

Mark point S so that FS = QR. Then, APQR = AXYS by SAS.

Since corresponding parts of congruent triangles are congruent,

PR = XS. It is given that PR = XZ, so XS = XZ by the Transitive

Property of Congruence. By the Isosceles Triangle Theorem,AS = AZ, so AXYS = AXYZ by AAS. Therefore, APQR = AXYZby the Transitive Property of Congruence.

Key Concept Conditions for HL Theorem

To use the HL Theorem, the triangles must meet three conditions.

Conditions

• There are two right triangles.

• The triangles have congruent hypotenuses.

• There is one pair of congruent legs.

Lesson 4-6 Congruence in Right Triangles 259

Proof Problem 1 Using the HL Theorem

On the basketball backboard brackets shown below, Z.ADC and /LBDC are

right angles and AC = BC. Are AADC and ABDC congruent? Explain.

How can you

visualize the two

right triangles?Imagine cutting AABCalong On eitherside of the cut, you gettriangles with the sameleg DC.

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• You are given that AADC and ABDC are right angles.So, AADC and ABDC are right triangles.

• The hypotenuses of the two right triangles are AC and BC.

You are given that AC = BC.

• DC is a common leg of both AADC and ABDC. DC = DC by the Reflexive Propertyof Congruence.

Yes, AADC = ABDC by the HL Theorem.

Got It? 1. a. Given: APRS and ARPQ are

right angles, BP = QR

Prove: APRS = ARPQ S

b. Reasoning Your friend says, "Suppose you have two right triangles withcongruent hypotenuses and one pair of congruent legs. It does not matter

which leg in the first triangle is congruent to which leg in the second

triangle. The triangles will be congruent." Is your friend correct? Explain.

260 Chapter 4 Congruent Triangles

Proof

How can you get

started?

Identify the hypotenuseof each right triangle.Prove that the

hypotenuses arecongruent.

Problem 2 Writing a Proof Using the HL Theorem

Given: bisects AD at C,

AB 1 EC, AB = W

Prove: AABC = ADEC

BE bisects AD.

Given

AS

DE 1 EC

Given

Z./AeC and

/.DEC are

right A.

Def. of 1 lines

AC = DC

Def. of bisector

A/teCand ADEC

are right A.

Def. of right triangle

AB = DE

Given

^ jjj Got It? 2. Given: CD = £-4, ̂ D is the perpendicularbisector of C£

Prove: ACBD s ABBA

AABC = ADEC

HL Theorem

C

BD

Lesson Check

Do you know HOW?

Are the two triangles congruent? If so, write the

congruence statement.

Do you UNDERSTAND?

5. Vocabulary A right triangle has side lengths of

5 cm, 12 cm, and 13 cm. What is the length of the

hypotenuse? How do you know?

6. Compare and Contrast How do the HL Theorem and

the SAS Postulate compare? How are they different?

Explain.

7. Error Analysis Your classmate says that there is not

enough information to determine whether the two

triangles below are congruent. Is your classmate

correct? Explain.

M

K

c PowerGeometry.com I Lesson 4-6 Congruence in Right Triangles 261

Practice and Problem-Solving Exercises

Practice 8. Developing Proof Complete the flow proof.

MATHEMATICAL

PRACTICES

Apply

Given: PS = Pf, /LPRS = APRT

Prove: APRS = APRT

jLPRS and ̂ PRTare =.

Given

^PffSand ̂ PRT

are supplementary.

A that form a linear

pair are supplementary.

LPRS and /.PRT

are right A.

ps^Wr

c.j_

' ̂= PR

d. ?

j ̂ See Problem 1.

APRS and APRT

are right A.

b. ?

APRS = APRT

e. ?

9. Developing Proof Complete the paragraph proof.

Given: AA and AD are right angles, AB = DE

Prove: AABE = ADEB

Proof: It is given that AA and AD are right angles. So, a. J_ by thedefinition of right triangles, b. because of the Reflexive Property of

Congruence. It is also given that C. ? .So, AABE = ADEB by d. ? .

L

n

ia Given: HV l GT, GH = TV,Proof / jg midpoint of HV

Prove: AIGH = AITV

11. Given: PM = Rf, ^ See Problem 2.Proof Pf ITf.mr ITf,

Mis the midpoint of TJ

Prove: APTM = ARMJ

H

V

Algebra For what values ofa; andy are the triangles congruent by HL?

12. 13. 3y + X

y+1y-x

x + 5

y+5

14. Study Exercise 8. Can you prove that APRS = APRT without using the HL

Theorem? Explain.

262 Chapter 4 Congruent Triangles

15. Think About a Plan AABC and APQR are

right triangular sections of a fire escape,

as shown. Is each story of the building the

same height? Explain.

• What can you tell from the diagram?

• How can you use congruent triangles here?

16. Writing "A HA!" exclaims your classmate.

"There must be an HA Theorem, sort of

like the HL Theorem!" Is your classmate

correct? Explain.

17. Given: RS = TU, RS 1 ST, TU 1 UV,Proof j js midpoint of RV

Prove: ARST^ ATUV

R

18. Given: ALNP is isosceles with base NP,Proof MN 1PL,ML = '^

Prove: AMNL = AQPL

L

-] T

UV

Constructions Copy the triangle and construct a triangle congruentto it using the given method.

20. HL19. SAS

21. ASA

23. Given: AGKE is isosceles with base G£,Proof angles, and

K is the midpoint of LD.

Prove: LG = DE

22. SSS

Given: LO bisects AMLN,Proof ^ W 1 In

Prove: ALMO = ALNO

M

25. Reasoning Are the triangles congruent? Explain.

C PowerGeo metiy.com

D

Lesson 4-6 Congruence in Right Triangles 263

\

26. a. Coordinate Geometry Graph the points y4(-5, 6), B(l, 3), -D(-8,0), and E{~2, -3).Draw AB, AE, ED, and DE. Label point C, the intersection of AE and BD.

b. Find the slopes of AE and BD. How would you describe Z-ACB and Z.ECD1

c. Algebra Write equations for AE and BD. What are the coordinates of C?

d. Use the Distance Formula to find AB, BC, DC, and DE.

e. Write a paragraph to prove that AABC = AEDC.

Challenge Geometry in 3 Dimensions For Exercises 27 and 28, use the hgure at the right.

27j Given: BE _L EA, BE L EC, AABC is equilateral

Prove: AAEB = ACEB

28. Given: AAEB = ACEB, M LEA,M EEC

Can you prove that AABC is equilateral? Explain.

SAT/Aa

Standardized Test Prep

29. You often walk your dog around the neighborhood.

Based on the diagram at the right, which of the

following statements about distances is true?

CK>SH=LH C^SH>LH

cj:>ph=ch c5:> bh<ch

School (5)

Park (P)

Cafe (C)

Short

Response

Library (L)

30. What is the midpoint of LM with endpoints L(2,7) and M{5, -1)?

CE> (3.5,3) CH^ (2,4.5)

CI5(3.5,4) CD (7, 6)

31. In equilateral AXYZ, name four pairs of congruent righttriangles. Explain why they are congruent.

Home (H)

Mixed Review"N

For Exercises 32 and 33, what type of triangle must ASTU be? Explain.

32. ASTU s AUTS 33. ASTU = AUST

4lb See Lesson 4-5.

Get Ready! To prepare for Lesson 4-7, do Exercises 34-36.

Can you conclude that the triangles are congruent? Explain.

34. AABC and ALMN 35. ALMN and AHJK

RN

^ See Lessons 4-3 and 4-6.

36. ARST and AABC

L J

K

MH

264 Chapter 4 Congruent Triangles