Page 1 of 7
5.4
Goal
Use the HL Congruence
Theorem and summarize
congruence postulates
and theorems.
Key Words
• hypotenuse p. 192
• leg of a right triangle
p. 192
Hypotenuse-Leg
Congruence Theorem: HL
The triangles that make up the skateboard ramp below
are right triangles.
The hypotenuse and a leg of one triangle
are congruent to the hypotenuse and
a leg of the other triangle.
hyp
ote
nu
se
use
ten
o
p
hy
leg
Student Help
VOCABULARY TIP
Remember that the
longest side of a right
triangle is called the
hypotenuse.
hypotenuse
leg
leg
THEOREM 5.2
Hypotenuse-Leg Congruence Theorem (HL)
Words
If the hypotenuse and a leg of a right triangle are congruent
to the hypotenuse and a leg of a second right triangle, then
the two triangles are congruent.
Symbols
leg
If TABC and TDEF are
right triangles, and
H
AC
&* c DF
&*, and
L
BC
&* c EF
&*,
then TABC c TDEF.
EXAMPLE
1
A
B
D
C
E
F
Determine When To Use HL
Is it possible to show that TJGH c THKJ
using the HL Congruence Theorem? Explain
your reasoning.
J
K
G
H
Solution
In the diagram, you are given that TJGH and THKJ are right triangles.
&* c JH
&* (hypotenuse) and you
By the Reflexive Property, you know JH
&
*
&**
are given that JG c HK (leg). You can use the HL Congruence Theorem
to show that TJGH c THKJ.
5.4
Hypotenuse-Leg Congruence Theorem: HL
257
Page 2 of 7
IStudent Help
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MORE EXAMPLES
More examples at
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EXAMPLE
2
Use the HL Congruence Theorem
Use the diagram to prove that TPRQ c TPRS.
S
Solution
Given
PR
&* ∏ SQ
&*
R
P
&* c PS
&*
PQ
Prove
P
TPRQ c TPRS
Statements
Reasons
&* ∏ SQ
&*
1. PR
1. Given
2. aPRQ and aPRS are right A.
2. ∏ lines form right angles.
3. T PRQ and T PRS are right
3. Definition of right triangle
triangles.
&* c PS
&*
H 4. PQ
4. Given
&* c PR
&*
L 5. PR
5. Reflexive Prop. of Congruence
6. T PRQ c T PRS
SUMMARY
6. HL Congruence Theorem
TRIANGLE CONGRUENCE POSTULATES AND THEOREMS
You have studied five ways to prove that TABC c TDEF.
SSS
SAS
ASA
AAS
HL
258
Chapter 5
Congruent Triangles
Side
AB
&* c DE
&*
Side
AC
&* c DF
&*
Side
BC
&* c EF
&*
Side
AB
&* c DE
&*
Angle
aB c aE
Side
BC
&* c EF
&*
Angle
aA c aD
Side
AB
&* c DE
&*
Angle
aB c aE
Angle
aA c aD
Angle
aB c aE
Side
BC
&* c EF
&*
B
A
AB
&* c DE
&*
Leg
BC
&* c EF
&*
C
D
B
A
C
A
D
C
D
F
E
C
D
B
A
F
E
B
A
F
E
B
TABC and TDEF are right triangles.
Hypotenuse
E
C
F
E
D
F
Page 3 of 7
Decide Whether Triangles are Congruent
3
EXAMPLE
Does the diagram give enough information to show that the triangles
are congruent? If so, state the postulate or theorem you would use.
a.
F
b.
B
C
G
A
Student Help
STUDY TIP
There is no SSA
Congruence Theorem
or Postulate, so you
cannot conclude that
the triangles in
Example 3(b) are
congruent.
E
H
D
Solution
a. From the diagram, you know that aBAC c aDAC, aB c aD, and
&* c DC
&**. You can use the AAS Congruence Theorem to show that
BC
TBAC c TDAC.
&* c HG
&**, EG
&* c EG
&*, and
b. From the diagram, you know that FG
aEFG c aEHG. Because the congruent angles are not included
between the congruent sides, you cannot show that TFGE c THGE.
4
EXAMPLE
Prove Triangles are Congruent
Use the information in the diagram
to prove that TRST c TUVW.
T
R
W
S U
V
Solution
Statements
Reasons
A 1. aS c aV
1. Given
&* c VW
&**
S 2. ST
2. Given
3. TUVW is equilateral.
3. Definition of equilateral triangle
4. aV c aW
4. Equilateral triangles are equiangular.
5. aT c aV
5. Given
A 6. aT c aW
6. Transitive Prop. of Congruence
7. TRST c TUVW
7. ASA Congruence Postulate
Decide Whether Triangles are Congruent
Does the diagram give enough information to show that the triangles
are congruent? If so, state the postulate or theorem you would use.
1. B
A
2.
C
R
S
3.
J
K
M
L
E
D
U
5.4
T
Hypotenuse-Leg Congruence Theorem: HL
259
Page 4 of 7
5.4 Exercises
Guided Practice
Vocabulary Check
Tell whether the segment is a leg or the hypotenuse of the
right triangle.
&*
1. AC
&*
2. BC
B
&*
3. AB
Skill Check
&*
4. KL
A
C
K
L
&*
5. KJ
6. JL
&*
J
Determine whether you are given enough information to show
that the triangles are congruent. Explain your answer.
7. D
8. M
E
P
9.
U
V
S
G
F
W
P
N
R
T
Practice and Applications
Extra Practice
HL Congruence Theorem Determine whether you can use the
HL Congruence Theorem to show that the triangles are congruent.
Explain your reasoning.
See p. 683.
10. A
D
E
11. J
12.
P
P
M
T
B
C
F
K
L
R
S
Landscaping To support a tree, you attach wires from the trunk
of the tree to stakes in the ground as shown below.
13. What information do you
need to know in order to use
the HL Congruence Theorem
to show that TJKL c TMKL?
Homework Help
Example 1: Exs. 10–13,
24, 29–31
Example 2: Ex. 32
Example 3: Exs. 13–24,
29–31
Example 4: Ex. 32
260
Chapter 5
14. Suppose K is the midpoint
L
&*. Name a theorem or
of JM
postulate you could use to
show that TJKL c TMKL.
Explain your reasoning.
Congruent Triangles
J
K
M
Page 5 of 7
You be the Judge Decide whether enough information is given
to show that the triangles are congruent. If so, state the theorem or
postulate you would use. Explain your reasoning.
15.
A
C
16. J
17.
K
Y
B
D
F
E
M
19. S
P
L
X
Z
F
18.
L P
T
V
N R
A
20.
B
C
E
H
G
21. A
22.
D
L
D
M
C
F
E
A
B
diagram shown at the right and asked which
congruence postulate or theorem can be used to
show that TABC c TCDA. Explain why all three
answers are correct.
D
C
Angie
Keith
TABC c TCDA
by the SSS
Congruence
Postulate.
HOMEWORK HELP
Extra help with problem
solving in Exs. 25–28 is
at classzone.com
23.
24. Logical Reasoning Three students are given the
Meghan
ICLASSZONE.COM
K
J
B
C
IStudent Help
D
U
J
TABC c TCDA by
the SAS Congruence
Postulate.
TABC c TCDA
by the
Hypotenuse-Leg
Congruence
Theorem.
Use the given information to sketch TLMN and TSTU.
Mark the triangles with the given information.
Visualize It!
25. aLNM and aTUS are right
&** c TS
&*, &**
&*
TU c LN
angles. LM
&** ∏ MN
&&, ST
&* ∏ &**
27. LM
TU ,
&** ∏ MN
&&, ST
&* ∏ &**
26. LM
TU ,
&** c ST
&*, LN
&* c SU
&*
LM
&* ∏ LN
&*, TS
&* ∏ SU
&*
28. ML
&** c NM
&& c UT
&** c ST
&*
LM
&* c SU
&*, MN
&** c TU
&**
LN
5.4
Hypotenuse-Leg Congruence Theorem: HL
261
Page 6 of 7
Missing Information What congruence is needed to show that the
triangles are congruent? Using that congruence, tell which theorem or
postulate you would use to show that the triangles are congruent.
F
29.
P
30. P
W
31.
V
G
H
J
X
Y
K
L
S
Z
R
32. Logical Reasoning Fill in the missing statements and reasons.
Given
BD
&* c FD
&*
B
F
&*.
D is the midpoint of CE
aBCD and aFED are right angles.
Prove TBCD c TFED
C
D
Statements
Reasons
&* c FD
&*
1. BD
1. _________?_________
2. _________?_________
2. Given
3. _________?_________
3. Definition of midpoint
E
4. aBCD and aFED are
4. _________?_________
right angles.
5. _________?_________ are
5. Definition of right triangle
right triangles.
6. TBCD c TFED
Standardized Test
Practice
6. _________?_________
33. Multi-Step Problem The diagram below is a plan showing the
light created by two spotlights. Both spotlights are the same
distance from the stage.
lights
B
stage
E
C
A
D
F
G
a. Show that TABD c TCBD. Tell what theorem or postulate you
use and explain your reasoning.
b. Is there another way to show that TABD c TCBD ? If so, tell how.
Explain your reasoning.
c. Are all four right triangles in the diagram congruent? Explain your
reasoning.
262
Chapter 5
Congruent Triangles
Page 7 of 7
Mixed Review
Parallel Lines Find ma1 and ma2. Explain your reasoning.
(Lesson 3.4)
34.
35.
36.
110
1 2
1 57
1
82
2
2
Showing Congruence Decide whether enough information is given
to show that the triangles are congruent. If so, state the theorem or
postulate you would use. Explain your reasoning. (Lessons 5.2, 5.3)
37.
38.
A
B
P
P
39. J
M
C
K
D
S
E
Algebra Skills
L
R
N
F
Evaluating Expressions Evaluate. (Skills Review, p. 670)
40. 2 p 4 5
41. 10 5 p 2
42. 3 42 11
43. 7 p 2 6 p 3
44. 3 p 5 2 p 7
45. 52 10 p 2
Quiz 2
Tell whether the theorem or postulate can be used to show
that TLMN c TQMP. (Lessons 5.3, 5.4)
1. ASA
2. AAS
3. HL
4. SSS
N
P
P
M
L
Tell whether enough information is given to show that the triangles
are congruent. If so, tell which theorem or postulate you would use.
Explain your reasoning. (Lessons 5.3, 5.4)
5. A
6.
B
D
J
P
C
L
8. E
F
9.
U
V
P
S
R
T
H
T
K
R
S
7.
J
10.
M
K
L
G
V
5.4
U
Hypotenuse-Leg Congruence Theorem: HL
263