Question

1. in the accompanying diagram, \\( \overline { h k } \\) bisects \\( \overline { i l } \\) and \\( \angle h \cong \angle k \\).

what is the most direct method of proof that could be used to prove \\( \triangle h i j \cong \triangle k l j \\)?
(1) \\( h l \cong h l \\)
(2) \\( s a s \cong s a s \\)
(3) \\( a a s \cong a a s \\)
(4) \\( a s a \cong a s a \\)

Explanation:

Step1: Analyze the given information

Since \( \overline{HK} \) bisects \( \overline{IL} \), we have \( IJ = LJ \). Also given \( \angle H=\angle K \). And \( \angle HJH=\angle KJL \) (vertical angles are equal).

Step2: Recall triangle congruence criteria

The AAS (Angle - Angle - Side) congruence criterion states that if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the two triangles are congruent.
In \( \triangle HIJ \) and \( \triangle KLJ \), we have \( \angle H=\angle K \), \( \angle HJH=\angle KJL \), and \( IJ = LJ \) (non - included side).

Answer:

(3) \( AAS\cong AAS \)