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.

hypotenuseleg

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?

5

3

5

3

5

3l

m n

o

p

q

r s

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 triangles2. The triangles have congruent hypotenuses3. 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

Statements Reasons

ab

c

d

e

Using the HL Theorem

Given: WJ congruent KZ and <W and <K are right angles.

Prove: Triangle JWZ congruent Triangle ZKJ

Statements Reasonsw

j

z

k

Using the HL Theorem

Given <PRS and <RPQ are right angles. SP congruent QR.

Prove: Triangle PRS congruent Triangle RPQ

Statements Reasons

s r

p q