4-6 Congruence in Right Triangles

4-6

4-6

Congruence in Right Triangles

1. Plan

What You¡¯ll Learn

Check Skills You¡¯ll Need

? To prove triangles

Tell whether the abbreviation identi?es a congruence statement.

congruent using the HL

Theorem

GO for Help

Lessons 4-2 and 4-3

Objectives

1

To prove triangles congruent

using the HL Theorem

1. SSS yes

2. SAS yes

3. SSA no

Examples

. . . And Why

4. ASA yes

5. AAS yes

6. AAA no

To show that one pattern can

be used to cut the fabric for

the two entrance flaps of a

tent, as in Example 1

Can you conclude that the two triangles are congruent? Explain.

7.

8.

yes; SAS

1

2

3

Math Background

yes; SAS

New Vocabulary ? hypotenuse ? legs of a right triangle

1

The Hypotenuse-Leg Theorem

In a right triangle, the side opposite the right angle

is the longest side and is called the hypotenuse.

The other two sides are called legs.

Hypotenuse

Theorem 4-6

The HL Theorem is an example

of using a SSA relationship to

prove triangles congruent. This

is not generally possible. The HL

Theorem also can be proved by

first proving the Pythagorean

Theorem and then applying it

to establish SSS congruence.

More Math Background: p. 196D

Legs

Right triangles provide a special case for which there

is an SSA congruence rule. (See Lesson 4-3, Exercise 32.)

It occurs when hypotenuses are congruent and one pair of legs are congruent.

Key Concepts

Real-World Connection

Using the HL Theorem

Using the HL Theorem

Lesson Planning and

Resources

See p. 196E for a list of the

resources that support this lesson.

Hypotenuse-Leg (HL) 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.

PowerPoint

Proof

Bell Ringer Practice

Proof of the HL Theorem

Given: #PQR and #XYZ are right triangles,

with right angles Q and Y respectively.

PR > XZ, and PQ > XY.

Prove: #PQR > #XYZ

P

X

Q

R Y

)

Proof: On #XYZ at the right, draw ZY.

Mark point S as shown so that YS = QR. Then,

#PQR > #XYS by SAS. By CPCTC, PR > XS.

It is given that PR > XZ, so XS > XZ by

the Transitive Property of Congruence.

Check Skills You¡¯ll Need

For intervention, direct students to:

Z

X

Using the SSS and SAS Postulates

Lesson 4-2: Examples 1 and 2

Extra Skills, Word Problems, Proof

Practice, Ch. 4

Using the ASA Postulate

S

Y

Z

Lesson 4-3: Example 1

Extra Skills, Word Problems, Proof

Practice, Ch. 4

By the Isosceles Triangle Theorem, &S > &Z, so #XYS > #XYZ by AAS.

Therefore, #PQR > #XYZ by the Transitive Property of Congruence.

Lesson 4-6 Congruence in Right Triangles

Special Needs

Below Level

L1

Point out that although only two letters are used to

name the HL Theorem, there are three conditions:

two right angles, one pair of congruent hypotenuses,

and one pair of congruent legs.

learning style: verbal

235

L2

Have students use the diagram in the proof of the

HL Theorem to explain why the HL Theorem is not a

special case of the SAS Postulate.

learning style: verbal

235

2. Teach

To use the HL Theorem, you must show that three conditions are met.

? There are two right triangles.

? The triangles have congruent hypotenuses.

? There is one pair of congruent legs.

Guided Instruction

1

Teaching Tip

Before reading the proof of the

HL Theorem, discuss a Plan for

Proof with the class. Draw XYZ,

and discuss why you might want

to extend ZY to form another

right angle. Make sure that the

class understands that) point S

can be located on ZY so that

YS = QR. This may seem like an

arbitrary construction, but careful

consideration of the subsequent

triangle congruence statements

will help students appreciate its

usefulness.

1

EXAMPLE

EXAMPLE

Real-World

Check whether the two right triangles meet the three conditions for the HL

Theorem.

? You are given that &CPA and

&MPA are right angles. #CPA

and #MPA are right triangles.

For: Right Triangles Activity

Use: Interactive Textbook, 4-6

? The hypotenuses of the

triangles are CA and MA. You

are given that CA > MA.

A

? PA is a leg of both #CPA and

#MPA. PA > PA by the

Re?exive Property of

Congruence.

Math Tip

#CPA > #MPA by the HL

Theorem. The triangles are the

same shape and size. You can use

one pattern for both ?aps.

Quick Check

Visual Learners

C

P

M

L

1 Which two triangles are congruent by the

HL Theorem? Write a correct congruence

statement. kLMN O kOQP

3

M

5

Proof

2

EXAMPLE

5

S

5

P

T

Using the HL Theorem

Prove: #CBD > #EBA

C

A

B

D

E

Proof:

CBD and EBA

s

are right .

Def. of  lines

Quick Check

2. CB O EB and mlCBD ¡Ù

mlEBA because AD is the

 of CE. It is given that

CD O EA. kCBD O kEBA

by HL.

3

Q

Given: CD > EA, AD is the perpendicular

bisector of CE.

AD is the

 bisector of CE.

Given

R

O

3

N

Highlight how three statements

come together in the conclusion

of the flow proof. Discuss how

this is similar to the way triangles

are proved congruent using SSS,

SAS, ASA, or AAS. Point out that

the flow proof uses the three

bulleted statements just before

Example 2.

CBD and EBA

are right triangles.

Def. of right triangle

CBD  EBA

HL Theorem

CB  EB

Def. of  bisector; def. of midpoint

CD  EA

Given

Quick Check

236

2 Prove that the two triangles you named in Quick Check 1 are congruent.

See margin.

Chapter 4 Congruent Triangles

Advanced Learners

English Language Learners ELL

L4

After completing Example 3, have students prove that

WZKJ must contain four right angles.

236

Connection

Tent Design On the tent, &CPA and &MPA are right angles and CA > MA.

Can you use one pattern to cut fabric for both ?aps of the tent? Explain.

Point out that applying the

Transitive Property of Congruence

to triangles is an extension of the

same property for segments and

angles.

2

EXAMPLE

learning style: verbal

Some students think right triangles can only use the

HL Theorem. Clarify that they can also apply the SSS,

SAS, and ASA Postulates and the AAS Theorem to

right triangles.

learning style: verbal

Proof

3

3

Using the HL Theorem

EXAMPLE

W

Z

J

K

Given: WJ > KZ, &W and &K are right angles.

Prove: #JWZ > #ZKJ

Statements

1.

2.

3.

4.

5.

Quick Check

Given

De?nition of right triangle

Re?exive Property of Congruence

Given

HL Theorem

P

3 Given: &PRS and &RPQ are right angles,

Q

SP > QR.

S

R

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

Practice and Problem Solving

Practice by Example

Example 1

GO for

Help

(page 236)

1¨C2. See left.

Write a short paragraph to explain why the two triangles are congruent.

L

A

N

1.

2.

F

10

M

10

D

6

C

1. kABC O kDEF by HL.

Both > are rt. > , AC

O DF , and CB O FE.

2. kLMP O kOMN by HL.

Both > are rt. >

because vert. ' are O;

LP O NO, and LM O OM.

Example 2

(page 236)

B

6

As students read the proof, ask:

Why is step 2 included in the

proof? It establishes a needed

condition for the HL Thm. to apply.

PowerPoint

1.

2.

3.

4.

5.

Prove: #PRS > #RPQ See back of book.

A

Teaching Tip

Reasons

&W and &K are right angles.

#JWZ and #ZKJ are right triangles.

JZ > JZ

WJ > KZ

#JWZ > #ZKJ

EXERCISES

EXAMPLE

P

E

Additional Examples

1 In Example 1, one student

wrote ¡°CPA  MPA by SAS.¡± Is

the student correct? Explain. No;

the congruent angles are not

included angles.

2 XYZ is isosceles. From vertex

X, a perpendicular is drawn to YZ ,

intersecting YZ at point M. Explain

why XMY  XMZ. kXMY

and kXMZ are right triangles,

XY ¡Ù XZ by def. of isosceles,

and XM O XM by Reflexive Prop.,

so kXMY O kXMZ by HL Thm.

3 Write a two-column proof.

Given: &ABC and &DCB are right

angles, AC  DB.

Prove: ABC  DCB

O

What additional information do you need to prove the triangles congruent by HL?

3. #BLT and #RKQ

lT and lQ are rt. '.

B

Q

R

K

L

4. #XRV and #TRV RX O RT or

XV O TV

R

X

T

T

V

5. Developing Proof Complete the ?ow proof.

S

Given: PS > PT, &PRS > &PRT

R

T

Prove: #PRS > #PRT

P

PRS and PRT are 

s

and supplementary .

Given (by diagram)

PRS and PRT

s

are right .

a. 9 O supp. '

are rt. '.

PS  PT

c. 9 Given

1. lABC and lDCB are rt.

angles. (Given)

2. kABC and kDCB are rt.

triangles. (Def. of rt. triangle)

3. AC O DB (Given)

4. BC O CB (Reflexive Prop.

of O)

5. kABC O kDCB (HL Thm.)

PRS and PRT

s

are right .

b. 9

Def. of

rt. k

PRS  PRT

e. 9 HL

PR  PR

d. 9 Re?exive Prop. of O

Lesson 4-6 Congruence in Right Triangles

237

Resources

? Daily Notetaking Guide 4-6 L3

? Daily Notetaking Guide 4-6¡ª

L1

Adapted Instruction

Closure

How are SAS and HL alike, and

how are they different? Both

prove triangles congruent using

two pairs of sides and one pair of

angles. SAS is a postulate, and

the angle is an included angle.

HL is a theorem, the triangle

must be right, and the angle is

not an included angle.

237

Proof

3. Practice

6. Given: AD > CB, &D and &B are right angles.

7. Developing Proof Complete the

two-column proof.

Example 3

1 A B 1-24

(page 237)

C Challenge

25-26

Test Prep

Mixed Review

27-30

31-39

J

K

Statements

Reasons

JL ' LM and LJ ' JK

&JLM and &LJK are right angles.

9 kMLJ and kKJL are rt. > .

MJ > KL

9 LJ O LJ

1.

2.

c.

4.

e.

Connection to Algebra

9 Given

9 Def. of #

De?nition of a right triangle

9 Given

Re?exive Property of

Congruence

f. 9 HL

a.

b.

3.

d.

5.

6. #JLM > #LJK

Exercises 10, 11 Students must

Proof

solve a system of two equations.

If necessary, have them reread the

Algebra Review on page 234.

8. Given: HV ' GT, GH > TV,

I is the midpoint of HV.

G

H

T

Exercise 14 Students may need to

B

copy the diagram and extend PM

to see that it is a transversal for

the parallel lines in the diagram.

Apply Your Skills

Exercises 16¨C19 Students will

need compasses and straightedges.

Have students demonstrate and

explain their constructions to

partners.

9. Antiques To repair an antique clock, a 12-toothed

wheel has to be made by cutting right triangles out

of a regular polygon that has twelve 4-cm sides. The

hypotenuse of each triangle is a side of the regular

polygon, and the shorter leg is 1 cm long. Explain

why the 12 triangles must be congruent.

HL; each rt. k has a O hyp. and side.

x 2 Algebra In Exercises 10 and 11, for what values of

x and y are the triangles congruent by HL?

10.

GPS x

GPS Guided Problem Solving

L3

L4

Enrichment

L2

Reteaching

L1

Adapted Practice

Class

L3

Date

Practice 4-6

Congruence in Right Triangles

AAS.

1. Given: AB # BC , ED # FE , AB  ED , AC  FD 2. Given: P and R are right angles, PS  QR

Prove: ABC  DEF

Prove: PQS  RSQ

E

C

P

Q

S

R

S

GO

D

Write a ?ow proof.

3. Given: MJ # NK , MN  MK

Prove: MJN  MJK

4. Given: GI  JI , GHI  JHI

Prove: IHG  IHJ

M

N

G

J

H

K

J

6. X

Y

7.

B

E

C

F

238

U

T

A

R

S

W

W

H

8.

9. L

M

G

11. Z

K

Z

P

N Q

B

10. S

T

V

U

R

J

W

A

X

12. C

D

Y

F

E

13. G

H

J

238

D

I

? Pearson Education, Inc. All rights reserved.

V

5.

y+1

3y

3y + x

y-x

x+5

Homework Help

T

M

nline

Visit:

Web Code: aue-0406

I

What additional information do you need to prove each pair of triangles

congruent by the HL Theorem?

11.

x+3

y+5

x ¡Ù ¨C1; y ¡Ù 3

x ¡Ù 3; y ¡Ù 2

12. Critical Thinking While working for a landscape architect, you are told to lay

Real-World

Connection

out a ?ower bed in the shape of a right triangle with sides of 3 yd and 7 yd.

Interest in antiques and shifts

Explain what else you need to know in order to make the ?ower bed.

in fashion have stabilized the

whether the 7-yd side is the hyp. or a leg

need for dial-clock repair skills.

Proof 13. Given: RS > TU, RS ' ST,

14. Given: JM > WP, JP 6 MW,

TU ' UV, T is the

JP ' PM

15. PS O PT so lS O lT

midpoint of RV.

Prove:

#JMP > #WPM

by the isosc. k thm.

Prove: #RST > #TUV

P

lPRS O lPRT.

J

R

kPRS O kPRT by

13¨C14. See back of book.

Write a two-column proof.

B

V

I

Prove: #IGH > #ITV

See back of book.

Visual Learners

F

C

L

Prove: # JLM > #LJK

To check students¡¯ understanding

of key skills and concepts, go over

Exercises 4, 7, 10, 14, 22.

A

D

M

Given: JL ' LM, LJ ' JK, MJ > KL

Homework Quick Check

Name

B

Prove: #ADC > #CBA See back of book.

Assignment Guide

Practice

A

U

Proof

W

V

15. Study Exercise 5. There is a different set of steps that will prove

#PRS > #PRT. Decide what they are. Then write a proof using these steps.

See left.

Chapter 4 Congruent Triangles

4. Assess & Reteach

Constructions Copy the triangle and construct a

triangle congruent to it using the method stated.

16. by SAS 16¨C19.

See back

18. by ASA of book.

17. by HL

PowerPoint

19. by SSS

20. 1. EB O DB; lA and

lC are rt. '. (Given) Proof 20. Given: EB > DB, &A and &C

2. kBEA and kBDC

are right angles, and

are rt. > . (Def. of rt. k)

B is the midpoint of AC.

3. B is the midpt. of AC.

Prove: #BEA > #BDC

See left.

(Given)

E

D

4. AB O BC (Def. of

midpt.)

5. kBEA O kBDC (HL)

A

B

C

Lesson Quiz

21. Given: LO bisects &MLN,

OM ' LM, and ON ' LN.

Prove: #LMO > #LNO

See margin.

M

For Exercises 1 and 2, tell

whether the HL Theorem can be

used to prove the triangles

congruent. If so, explain. If not,

write not possible.

O

1.

L

A

N

C

B

22. Open-Ended You are the DJ for the school dance. To set up, you have placed

one speaker in the corner of the platform. What measurement(s) could you

make with a tape measure to make sure that a matching speaker is in the other

corner at exactly the same angle? Explain why your method works.

See margin.

23. a. Coordinate Geometry Use grid paper. Graph the points E(-1, -1),

F(-2, -6), G(-4, -4), and D(-6, -2). Connect the points with segments.

b. Find the slope for each of DG, GF, and GE. a¨Cc. See back of book.

c. Use your answer to part (b) to describe &EGD and &EGF.

d. Use the Distance Formula to ?nd DE and FE. DE ¡Ù "26; FE ¡Ù "26

e. Write a paragraph to prove that #EGD > #EGF. See back of book.

Exercise 22

C

24. Critical Thinking ¡°A HA!¡± exclaims Francis. ¡°There is an HA Theorem . . . ,

something like the HL Theorem!¡± Explain what Francis is saying and why he is

correct or incorrect. An HA Thm. is the same as AAS with AAS corr. to

the rt. l, an acute l, and the hyp.

Geometry in 3 Dimensions Use the ?gure at the right for

Exercises 25 and 26.

B

Challenge

Proof

25. Given: BE ' EA, BE ' EC, #ABC is equilateral.

Prove: #AEB > #CEB See margin, p. 240.

26. Given: #AEB > #CEB, BE ' EA, and BE ' EC.

E

A

Can you prove that #ABC is equilateral? Explain.

No; AB O CB because kAEB O kCEB, but AC doesn¡¯t have

to be O to AB or to CB.

C

D

Yes; use congruent

hypotenuses and leg BC

to prove kABC O kDBC.

2.

R

Q

E

T

W

not possible

For Exercises 3 and 4, what

additional information do you

need to prove the triangles

congruent by the HL Theorem?

3. LMX  LOX

L

M

X

Test Prep

O

Multiple Choice

In Exercises 27 and 28, which additional congruence statement could you use

to prove that #BJK > #CFH by HL?

A

27. Given: BJ > CF A

J

F

A. JK > FH

B. &B > &C

C. AJ > AF

D. &BJK > &CFH

28. Given: BK > CH

F. JK > FH

B

K

H

N

LM O LO

4. AMD  CNB

A

N

B

C

H

G. &B > &C

H. JB > FC

J. &BJK > &CFH

D

M

C

AM O CN or MD O NB

lesson quiz, , Web Code: aua-0406

21. 1. LO bisects lMLN,

OM  LM, ON  LN,

(Given)

2. lM and lN are rt. '

(Def. of )

Lesson 4-6 Congruence in Right Triangles

3. lMLO O lNLO (Def.

of l bis.)

4. lM O lN (All rt. ' are

O.)

5. LO O LO (Reflexive

Prop. of O)

239

6. kLMO O kLNO (AAS)

22. Answers may vary.

Sample: Measure 2

sides of the k

formed by the amp.

and the platform¡¯s

corner. Since the >

will be O by HL or

SAS, the ' are the

same.

239

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