5.4 Hypotenuse-Leg Congruence Theorem: HL 257
Goal
Use the HL Congruence
Theorem and summarize
congruence postulates
and theorems.
Key Words
• hypotenuse p. 192
• leg of a right triangle
p. 192
5.4
5.4
Hypotenuse-Leg
Congruence Theorem: HL
Is it possible to show that TJGH c THKJ
using the HL Congruence Theorem? Explain
your reasoning.
Solution
In the diagram, you are given that TJGH and THKJ are right triangles.
By the Reflexive Property, you know JH
&*
c JH
&*
(hypotenuse) and you
are given that JG
&*
c HK
&**
(leg). You can use the HL Congruence Theorem
to show that TJGH c THKJ.
G
J K
H
EXAMPLE
1
Determine When To Use HL
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 If TABC and TDEF are
right triangles, and
H AC&* c DF&*, and
L BC&* c EF&*,
then TABC c TDEF.
THEOREM 5.2
A
B C
D
E F
V
OCABULARY
T
IP
Remember that the
longest side of a right
triangle is called the
hypotenuse.
hypotenuse
leg
leg
Student Help
The hypotenuse and a leg of one triangle
are congruent to the hypotenuse and
a leg of the other triangle.
leg
leg
hypotenuse
hypotenuse
The triangles that make up the skateboard ramp below
are right triangles.
Page 1 of 7
258 Chapter 5 Congruent Triangles
Use the diagram to prove that TPRQ c TPRS.
Solution
Given
©
PR
&*
∏ SQ
&*
PQ
&*
c PS
&*
Prove
©
TPRQ c TPRS
Statements Reasons
1. PR
&*
∏ SQ
&*
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.
H
4. PQ
&*
c PS
&*
4. Given
L
5. PR
&* c
PR
&*
5. Reflexive Prop. of Congruence
6. T PRQ c T PRS 6. HL Congruence Theorem
S
R
P
P
EXAMPLE
2
Use the HL Congruence Theorem
M
ORE
E
XAMPLES
More examples at
classzone
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IStudent Help
I CLA S S ZO N E .CO M
TRIANGLE CONGRUENCE POSTULATES AND THEOREMS
You have studied five ways to prove that TABC c TDEF.
SSS Side AB&* c DE&*
Side AC&* c DF&*
Side BC&* c EF&*
SAS Side AB&* c DE&*
Angle aB c aE
Side BC&* c EF&*
ASA Angle aA c aD
Side AB&* c DE&*
Angle aB c aE
AAS Angle aA c aD
Angle aB c aE
Side BC&* c EF&*
HL TABC and TDEF are right triangles.
Hypotenuse AB&* c DE&*
Leg BC&* c EF&*
A C
B
D F
E
A C
B
D F
E
A C
B
D F
E
A C
B
D F
E
A C
B
D F
E
SUMMARY
Page 2 of 7
5.4 Hypotenuse-Leg Congruence Theorem: HL 259
Does the diagram give enough information to show that the triangles
are congruent? If so, state the postulate or theorem you would use.
a. b.
Solution
a. From the diagram, you know that aBAC c aDAC, aB c aD, and
BC
&*
c DC
&**
. You can use the AAS Congruence Theorem to show that
TBAC c TDAC.
b. From the diagram, you know that FG
&*
c HG
&**
, EG
&*
c EG
&*
, and
aEFG c aEHG. Because the congruent angles are not included
between the congruent sides, you cannot show that TFGE c THGE.
G
F
E
H
A
B
D
C
EXAMPLE
3
Decide Whether Triangles are Congruent
S
TUDY
T
IP
There is no SSA
Congruence Theorem
or Postulate, so you
cannot conclude that
the triangles in
Example 3(b) are
congruent.
Student Help
Does the diagram give enough information to show that the triangles
are congruent? If so, state the postulate or theorem you would use.
1. 2. 3.
M
J K
L
U
R S
T
A
B
E
C
D
Decide Whether Triangles are Congruent
Use the information in the diagram
to prove that TRST c TUVW.
Solution
Statements Reasons
A 1. aS c aV 1. Given
S 2. ST
&*
c VW
&**
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
R S
T
U V
W
EXAMPLE
4
Prove Triangles are Congruent
Page 3 of 7
260 Chapter 5 Congruent Triangles
Tell whether the segment is a leg or the hypotenuse of the
right triangle.
1. AC
&*
4. KL
&*
2. BC
&*
5. KJ
&*
3. AB
&*
6. JL&*
Determine whether you are given enough information to show
that the triangles are congruent. Explain your answer.
7. 8. 9.
HL Congruence Theorem
Determine whether you can use the
HL Congruence Theorem to show that the triangles are congruent.
Explain your reasoning.
10. 11. 12.
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?
14. Suppose K is the midpoint
of JM
&*
. Name a theorem or
postulate you could use to
show that TJKL c TMKL.
Explain your reasoning.
P
R S
T
P
J
K L
M
A D E
B C F
Practice and Applications
S
R T
W
U V
M P
N
P
D
G F
E
Skill Check
Vocabulary Check
Guided Practice
K
J
L
B
A
C
Exercises
5.4
5.4
Extra Practice
See p. 683.
Example 1: Exs. 10–13,
24, 29–31
Example 2: Ex. 32
Example 3: Exs. 13–24,
29–31
Example 4: Ex. 32
Homework Help
J K M
L
J K M
L
Page 4 of 7
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. 16. 17.
18. 19. 20.
21. 22. 23.
24.
Logical Reasoning
Three students are given the
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.
Use the given information to sketch TLMN and TSTU.
Mark the triangles with the given information.
25. aLNM and aTUS are right 26. LM
&**
∏ MN
&&
, ST
&*
∏ TU
&**
,
angles. LM
&**
c TS
&*
, TU
&**
c LN
&*
LM
&**
c ST
&*
, LN
&*
c SU
&*
27. LM
&**
∏ MN
&&
, ST
&*
∏ TU
&**
, 28. ML
&*
∏ LN
&*
, TS
&*
∏ SU
&*
LM
&**
c NM
&&
c UT
&**
c ST
&*
LN
&*
c SU
&*
, MN
&**
c TU
&**
Visualize It!
D
A
C
B
C F
D
E
L
J
K
M
C
D B
A
BA
E D
C
T
U
S V
JG
F
H
N
L
R
P
M
P
J
Z
LK
XY
D
EF
A
BC
You be the Judge
5.4 Hypotenuse-Leg Congruence Theorem: HL 261
Meghan
TABC c TCDA
by the SSS
Congruence
Postulate.
Keith
TABC c TCDA by
the SAS Congruence
Postulate.
Angie
TABC c TCDA
by the
Hypotenuse-Leg
Congruence
Theorem.
H
OMEWORK
H
ELP
Extra help with problem
solving in Exs. 25–28 is
at
classzone
.com
IStudent Help
I CLA S S ZO N E .CO M
Page 5 of 7
262 Chapter 5 Congruent Triangles
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.
29. 30. 31.
32.
Logical Reasoning
Fill in the missing statements and reasons.
Given
©
BD
&*
c FD
&*
D is the midpoint of CE
&*
.
aBCD and aFED are right angles.
Prove
©
TBCD c TFED
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.
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.
Standardized Test
Practice
Statements Reasons
1. BD
&*
c FD
&*
1. _________?_________
2. _________?_________ 2. Given
3. _________?_________ 3. Definition of midpoint
4. aBCD and aFED are
right angles. 4. _________?_________
5. _________?_________ are 5. Definition of right triangle
right triangles.
6. TBCD c TFED 6. _________?_________
C D
B
E
F
Z
Y
X
V
W
S
P
R
P
K L
J
G H
F
A
B
C
D
E
F
G
lights
stage
Page 6 of 7
Parallel Lines
Find ma1 and ma2. Explain your reasoning.
(Lesson 3.4)
34. 35. 36.
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. 39.
Evaluating Expressions
Evaluate. (Skills Review, p. 670)
40. 2 p 4 1 5 41. 10 2 5 p 2 42. 3 1 4
2
2 11
43. 7 p 2 1 6 p 3 44. 3 p 5 2 2 p 7 45. 5
2
2 10 p 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
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. 6. 7.
8. 9. 10.
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V
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F
T U
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L K
J
P
D
A
C
B
Quiz 2
Algebra Skills
K L
J
N
M
S R
P
P
B C
A
E F
D
1
2
578
1
2
828
1
2
1108
Mixed Review
5.4 Hypotenuse-Leg Congruence Theorem: HL 263
P
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Page 7 of 7
