Question

complete the proof that ( \triangle hikcong\triangle gjk ).

statement	reason
2 ( overline{hk}congoverline{gk} )	given
3 ( angle gkjcongangle hki )	vertical angle theorem
4 ( \triangle hikcong\triangle gjk )

Explanation:

Step1: Recall the Side - Angle - Side (SAS) Congruence Postulate

The SAS postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Step2: Identify the corresponding parts in \(\triangle HIK\) and \(\triangle GJK\)

We are given that \(\overline{IK}\cong\overline{JK}\), \(\overline{HK}\cong\overline{GK}\), and \(\angle GKJ\cong\angle HKI\) (by the Vertical Angle Theorem). The angle \(\angle GKJ\) is included between \(\overline{GK}\) and \(\overline{JK}\) in \(\triangle GJK\), and the angle \(\angle HKI\) is included between \(\overline{HK}\) and \(\overline{IK}\) in \(\triangle HIK\).

Answer:

By the Side - Angle - Side (SAS) Congruence Postulate.