GE_MNLESE385795

Name Class Date

Explore

Is There a Side-Side-Angle

Congruence Theorem?

You have already seen several theorems for proving that triangles are congruent. In this

Explore, you will investigate whether there is a SSA Triangle Congruence Theorem.

Follow these steps to draw △ABC such that m∠A = 30°, AB = 6 cm, and BC = 4 cm.

The goal is to determine whether two side lengths and the measure of a non-included

angle (SSA) determine a unique triangle.

A

Use a protractor to draw a large 30° angle on a separate sheet of paper.

Label it ∠A.

B

Use a ruler to locate point B on one ray of ∠A so that AB = 6 cm.

C

Now draw

_

BC so that BC = 4 cm. To do this, open a compass to a

distance of 4 cm. Place the point of the compass on point B and draw

an arc. Plot point C where the arc intersects the side of ∠A. Draw

_

BC

to complete

△ABC.

D

What do you notice? Is it possible to draw only one △ABC with the

given side length? Explain.

Reflect

1. Do you think that SSA is sufficient to prove congruence? Why or why not?

2. Discussion Your friend said that there is a special case where SSA can be used to prove congruence.

Namely, when the non-included angle was a right angle. Is your friend right? Explain.

Resource

Locker

A

30°

B

A

30°

6 cm

B

A

C

30°

6 cm

4 cm

Module 6

295

Lesson 3

6.3 HL Triangle Congruence

Essential Question: What does the HL Triangle Congruence Theorem tell you about two

triangles?

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Explain 1

Justifying the Hypotenuse-Leg Congruence Theorem

In a right triangle, the side opposite the right angle is the hypotenuse.

The two sides that form the sides of the right angle are the legs.

You have learned four ways to prove that triangles are congruent.

• Angle-Side-Angle (ASA) Congruence Theorem • Side-Angle-Side (SAS) Congruence Theorem

• Side-Side-Side (SSS) Congruence Theorem • Angle-Angle-Side (AAS) Congruence Theorem

The Hypotenuse-Leg (HL) Triangle Congruence Theorem is a special case that

allows you to show that two right triangles are congruent.

Hypotenuse-Leg (HL) Triangle Congruence Theorem

If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a

leg of another right triangle, then the triangles are congruent.

Example 1 Prove the HL Triangle Congruence Theorem.

Given: △ABC and △DEF are right triangles;

∠C and ∠F are right angles.

_

AB ≅

_

DE and

_

BC ≅

_

EF

Prove: △ABC ≅ △DEF

By the Pythagorean Theorem, a

2

+ b

2

= c

2

and

2

+

2

= f

2

. It is given that

_

AB ≅

_

DE , so AB = DE and c = ƒ. Therefore, c

2

= f

2

and a

2

+ b

2

=

2

+

2

. It is given that

_

BC ≅

_

EF , so BC = EF and a = d. Substituting a for d in the above equation, a

2

+ b

2

=

2

+

2

.

Subtracting a

2

from each side shows that b

2

= ,

2

and taking the square root of each side, b = .

This shows that

_

AC ≅ .

Therefore, △ABC ≅ △DEF by .

Your Turn

3. Determine whether there is enough information to prove that

triangles △VWX and △YXW are congruent. Explain.

hypotenuse

legs

D

E

FA

B

C

d

e

f

a

b

c

XW

V

Z

Y

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Lesson 3

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Explain 2

Applying the HL Triangle Congruence Theorem

Example 2 Use the HL Congruence Theorem to prove that the triangles are

congruent.

A

Given: ∠P and ∠R are right angles.

_

PS ≅

_

RQ

Prove: △PQS ≅ △RSQ

Statements Reasons

1. ∠P and ∠R are right angles. 1. Given

2.

_

PS ≅

_

RQ

2. Given

3.

_

SQ ≅

_

SQ

3. Reflexive Property of Congruence

4. △

PQS ≅ △RSQ

4. HL Triangle Congruence Theorem

B

Given: ∠J and ∠L are right angles. K is the midpoint

of

_

JL and

_

MN .

Prove: △JKN ≅ △LKM

Statements Reasons

1. ∠J and ∠L are right angles. 1.

2. K is the midpoint of

_

JL and

_

MN . 2.

3.

_

JK ≅

_

LK and

_

MK ≅

_

NK

3.

4. △JKN ≅ △LKM 4.

Reflect

4. Is it possible to write the proof in Part B without using the HL Triangle Congruence Theorem? Explain.

Your Turn

Use the HL Congruence Theorem to prove that the triangles are congruent.

5. Given: ∠CAB and ∠DBA are right angles.

_

AD ≅

_

BC

Prove: △ABC ≅ △BAD

S

Q

R

P

LJ

M

K

N

B

DC

A

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Lesson 3

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Elaborate

6. You draw a right triangle with a hypotenuse that is 5 inches long. A friend also draws

a right triangle with a hypotenuse that is 5 inches long. Can you conclude that the

triangles are congruent using the HL Congruence Theorem? If not, what else would

you need to know in order to conclude that the triangles arecongruent?

7.

Essential Question Check-In How is the HL Triangle Congruence Theorem similar

to and different from the ASA, SAS, SSS, and AAS Triangle Congruence Theorems?

1. Tyrell used geometry software to construct

∠ABC so that m∠ABC = 20°. Then he dragged point A

so that AB = 6 cm. He used the software’s compass tool

to construct a circle centered at point A with radius 3 cm.

Based on this construction, is there a unique △ABC with

m∠ABC = 20°, AB = 6 cm, and AC = 3 cm? Explain.

Determine whether enough information is given to prove that the triangles are

congruent. Explain your answer.

2. △ABC and △DCB 3. △PQR and △STU

• Online Homework

• Hints and Help

• Extra Practice

Evaluate: Homework and Practice

C

B

D

A

T

S

R

Q

P

U

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4. △GKJ and △JHG 5. △EFG and △SQR

Write a two-column proof, using the HL Congruence Theorem, to prove that the

triangles are congruent.

6. Given: ∠A and ∠B are right angles.

―

AB ≅

―

DC

Prove: △ABC ≅ △DCB

7. Given: ∠FGH and ∠JHK are right angles.

H is the midpoint of

_

GK .

_

FH ≅

_

JK

Prove: △FGH ≅ △JHK

8. Given:

―

MP is perpendicular to

―

QR .

N is the midpoint of

―

MP .

_

QP ≅

―

RM

Prove: △MNR ≅ △PNQ

G H

JK

F

E G

Q

SR

B C

A D

G

H

K

J

F

P

R

N

Q

M

Statements Reasons

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Lesson 3

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9. Given: ∠ADC and ∠BDC are right angles.

―

AC ≅

―

BC

Prove:

―

AD ≅

―

BD

Algebra What value of x will make the given triangles congruent? Explain.

10. △JKL and △JKM 11. △ABC and △ABD

12. △STV and △UVT 13. △MPQ and △PMN

A

D

C

B

2x

+

2

5x

-

19

L

KJ

M

x

+

8

3x

-

14

A B

C

D

4x

+

2

6x

-

7

S T

UV

7x

-

5

4x

+

25

M

P

Q

N

Statements Reasons

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Algebra Use the HL Triangle Congruence Theorem to show that △ABC ≅ △DEF .

(Hint: Use the Distance Formula to show that appropriate sides are congruent. Use

the slope formula to show that appropriate angles are right angles.)

14.

15.

16.

Communicate Mathematical Ideas A vertical tower is supported

by two guy wires, as shown. The guy wires are both 58 feet long. Is it

possible to determine the distance from the bottom of guy wire

_

AB to

the bottom of the tower? If so, find the distance. If not, explain why not.

D

-

4

-

2

4

2

y

0

42

x

-

4

A

C

B

E

F

D

-

4

-

2

4

2

y

0

4

x

-

4

-

2

A

C

B

E

F

A

B C

D

34 ft

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Tarczyski/Panther Media/age fotostock

17. A carpenter built a truss, as shown, to support the roof of a

doghouse.

a. The carpenter knows that

_

KJ ≅

_

MJ . Can the carpenter conclude that

△KJL ≅ △MJL? Why or why not?

b.

What If? Suppose the carpenter also knows that ∠JLK is a right angle. Can the carpenter now

conclude that △KJL ≅ △MJL? Explain.

18.

Counterexamples Denise said that if two right triangles share

a common hypotenuse, then the triangles must be congruent.

Sketch a figure that serves as a counterexample to show that

Denise’s statement is not true.

19.

Multi-Step The front of a tent is covered by a

triangular flap of material. The figure represents the

front of the tent, with

_

PS ⟘

_

QR and

_

PQ ≅

_

PR . Jonah

needs to determine the perimeter of △PQR so that he

can replace the zipper on the tent. Find the perimeter.

Explain your steps.

J

L

K

M

P

S

Q

R

5 ft

4 ft

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20. A student is asked to write a two-column proof for the following.

Given: ∠ABC and ∠DCB are right angles.

_

AC ≅

_

BD

Prove:

_

AB ≅

_

DC

Assuming the student writes the proof correctly, which of the following

will appear as a statement or reason in the proof? Select all that apply.

A. ASA Triangle Congruence Theorem D. Reflexive Property of Congruence

B.

_

BC ≅

_

BC E. CPCTC

C. ∠A ≅ ∠D F. HL Triangle Congruence Theorem

H.O.T. Focus on Higher Order Thinking

21. Analyze Relationships Is it possible for a right triangle with a leg that is 10 inches long

and a hypotenuse that is 26 inches long to be congruent to a right triangle with a leg

that is 24 inches long and a hypotenuse that is 26 inches long? Explain.

22.

Communicate Mathematical Ideas In the figure,

_

JK ≅

_

LM ,

_

JM ≅

_

LK ,

and ∠J and ∠L are right angles. Describe how you could use three different

congruence theorems to prove that △JKM ≅ △LMK.

A

B

D

C

J K

M L

Statements Reasons

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23. Justify Reasoning Do you think there is an LL Triangle Congruence Theorem? That is, if the

legs of one right triangle are congruent to the legs of another right triangle, are the triangles

necessarily congruent? If so, write a proof of the theorem. If not, provide a counterexample.

Lesson Performance Task

The figure shows kite ABCD.

a. What would you need to know about the relationship

between

_

AC and

_

DB in order to prove that △ADE ≅ △ABE

and △CDE ≅ △CBE by the HL Triangle Congruence Theorem?

b. Can you prove that △ADC and △ABC are congruent

using the HL Triangle Congruence Theorem?

Explain why or why not.

c. How can you prove that the two triangles named in

Part b are in fact congruent, even without the

additional piece of information?

A

BD

E

C

Statements Reasons

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Lesson 3

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