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Congruence in Right Triangles
Academic Geometry
The HL 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
leg
leg
The HL Theorem
Right triangles provide a special case for
congruence. There is an SSA congruence rule.
It occurs when the hypotenuses are congruent
and one pair of legs are congruent.
Theorem 4-6
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.
The HL Theorem
Which two triangles are congruent by the HL
Theorem?
p
3
l
r
5
m
3
q
n
s
o
5
t
The HL Theorem
Are these triangles congruent using the HL
Theorem?
HL Theorem
To use the HL Theorem 3 conditions must be
met:
1. There are 2 right triangles
2. The triangles have congruent hypotenuses
3. There is one pair of congruent legs
Using the HL Theorem
Given: CD congruent EA, AD is the perpendicular
bisector of CE
Prove: Triangle CBD congruent Triangle EBA
c
Statements
Reasons
a
b
e
d
Using the HL Theorem
Given: WJ congruent KZ and <W and <K are right
angles.
Prove: Triangle JWZ congruent Triangle ZKJ
w
Statements
z
Reasons
j
k
Using the HL Theorem
Given <PRS and <RPQ are right angles. SP congruent
QR.
Prove: Triangle PRS congruent Triangle RPQ
p
Statements
q
Reasons
s
r