Question

k is the midpoint of \\( \overline { g i } \\) and \\( \overline { h j } \\) and \\( h i \cong g i \\). complete the proof that \\( \triangle h i k \cong \triangle j g k \\).

Explanation:

Step1: Determine the reason for $\overline{HK}\cong\overline{JK}$

Since \(K\) is the mid - point of \(\overline{HJ}\) (given in statement 2), by the definition of a mid - point (a mid - point of a line segment divides the line segment into two congruent parts), we have \(\overline{HK}\cong\overline{JK}\).

Step2: Determine the reason for \(\triangle HIK\cong\triangle JGK\)

We have \(\overline{HI}\cong\overline{GJ}\) (statement 3), \(\overline{GK}\cong\overline{IK}\) (statement 4), and \(\overline{HK}\cong\overline{JK}\) (statement 5).
By the Side - Side - Side (SSS) congruence criterion (if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent), \(\triangle HIK\cong\triangle JGK\).

Answer:

1. Definition of mid - point; 6. SSS (Side - Side - Side) congruence criterion.