In the accompanying diagram, HK bisects IL, and ∠H is congruent to ∠K.

Question

What is the most direct method of proof that could be used to prove that ΔHIJ is congruent to ΔKLJ?
AAS

HL

ASA

SAS

Answer 1

To prove that \( \triangle HIJ \) is congruent to \( \triangle KLJ \), we can analyze the information given:

1. \( HK \) bisects \( IL \).
2. \( \angle H \) is congruent to \( \angle K \).

Given that \( HK \) bisects \( IL \), we have that \( IH \) is congruent to \( KL \) by the definition of a bisector.

Now we can consider the triangles \( \triangle HIJ \) and \( \triangle KLJ \):

• We have \( \angle H \cong \angle K \).
• We also have \( IJ = LJ \) (the segments opposite the congruent angles).

This leads us to use the Angle-Angle-Side (AAS) criteria for triangle congruence since we have two angles and the side between them congruent in both triangles.

Thus, the most direct method of proof to show that \( \triangle HIJ \cong \triangle KLJ \) is through AAS.