Question

△ijk and △ghk are equilateral. complete the proof that △hik ≅ △gjk.

statement
1 △ijk is equilateral
2 △ghk is equilateral
3 (overline{ik}congoverline{jk})
4 (overline{gk}congoverline{hk})
5 (∠gkjcong∠hki)
6 (△hikcong△gjk)

reason
given
given
definition of equilateral triangle
definition of equilateral triangle

Explanation:

Step1: Vertical angles are congruent

Vertical angles are formed when two lines intersect. By the vertical - angles theorem, \(\angle GKJ\) and \(\angle HKI\) are vertical angles, so \(\angle GKJ\cong\angle HKI\).

Step2: Use the Side - Angle - Side (SAS) congruence criterion

We know that \(\overline{IK}\cong\overline{JK}\) (from step 3, definition of equilateral triangle), \(\overline{GK}\cong\overline{HK}\) (from step 4, definition of equilateral triangle), and \(\angle GKJ\cong\angle HKI\) (from step 5).
The Side - Angle - Side (SAS) congruence criterion 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.
For \(\triangle HIK\) and \(\triangle GJK\), in \(\triangle HIK\), we have sides \(\overline{HK}\) and \(\overline{IK}\) with included angle \(\angle HKI\), and in \(\triangle GJK\), we have sides \(\overline{GK}\) and \(\overline{JK}\) with included angle \(\angle GKJ\).

Answer:

1. Vertical - angles theorem; 6. SAS (Side - Angle - Side) congruence criterion.