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Geometry notes 5.4
Hypotenuse-Leg Congruence Theorem (HL)
Hypotenuse-Leg Congruence Theorem (HL): 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.
If
D
A
ABC and DEF are right triangles, and
AC and DF , and BC  EF , then
ABC ≅ DEF .
B
Using the HL Theorem:
J
K
G
H
C
E
F
Example:
Is it possible to show that JGH  HKJ using the HL Theorem? Explain your reasoning.
In the diagram, we are given that JGH and HKJ are right triangles.
By the Reflexive Property, we know JH  JH (hypotenuse) and we are given that JG  HK
(leg). We can use the HL Congruence Theorem to show that JGH  HKJ .
Example:
Use the diagram to prove that PRQ  PRS .
S
P
R
Given: PR  SQ and PQ  PS
Q
Prove: PRQ  PRS
Statement
Reason ____________________________
1. PR  SQ
1. Given
2. PRQ and PRS are right angles
2.  lines form rights angles
3.
PRQ and PRS are right 's
3. Def. of right triangle
4. PQ  PS
4. Given
5. PR  PR
6. PRQ  PRS
5. Reflexive Property
6. HL Congruence Theorem