The document presents two triangle congruence theorems: 1) The Hypotenuse-Leg (HyL) Congruence Theorem states that if the hypotenuse and a leg of one right triangle are congruent to the corresponding hypotenuse and leg of another triangle, then the triangles are congruent. 2) The Hypotenuse-Acute Angle (HyA) Congruence Theorem states that if the hypotenuse and an acute angle of one right triangle are congruent to the corresponding hypotenuse and acute angle of another right triangle, then the triangles are congruent. Proofs of sample triangle congruences are provided for each theorem.

3. HyL (Hypotenuse-Leg)
Congruence Theorem
If the hypotenuse and a leg of one right
triangle are congruent to the
corresponding hypotenuse and a leg of
another triangle, then the triangles are
congruent.

4. A
B C
D
Given:
AD  BC
BA  AC
Prove:
ABD  ACD

5. PROOF
Statement Reasons
Perpendiculars form right angles
A triangle with a 90 angle is a
right triangle
Given
ADB and ADC are right triangles
ADB and ADC are right triangles
AD = AD Reflexive
Hypotenuse-leg Postulate
BA  AC
ADB  ADC

6. HyA(Hypotenuse-Acute
angle) Congruence Theorem
If the hypotenuse and an acute angle of
one right triangle are congruent to the
corresponding hypotenuse and an acute
angle of another right triangle, then the
triangles are congruent

7. C and R are right angles
A
AB  PQ
B  Q
B C
P
R
Q
Given:
Prove:
ABC  PQR

8. PROOF
Statement Reasons
Given
Any two right
angles are
congruent
HyA Congruence
C and R are right angles
AB  PQ
B  Q
C  R
ABC  PQR Theorem