Question

in the diagram below, △gjh is an equilateral triangle, j is the midpoint of ik, and gk≅hi. complete the proof that △jih≅△jkg.

1. hi≅gk given
2. △gjh is an equilateral triangle given
3. jh≅jg definition of an equilateral triangle
4. j is the midpoint of ik given
5. ji≅jk
6. △jih≅△jkg

Explanation:

Step1: Recall mid - point property

If \(J\) is the mid - point of \(\overline{IK}\), by the definition of a mid - point, it divides the line segment \(\overline{IK}\) into two equal parts. So, \(\overline{JI}\cong\overline{JK}\).

Step2: Apply congruence criterion

In \(\triangle JIH\) and \(\triangle JKG\), we have \(\overline{HI}\cong\overline{GK}\) (given), \(\overline{JH}\cong\overline{JG}\) (from the property of equilateral triangle \(\triangle GJH\)) and \(\overline{JI}\cong\overline{JK}\) (from the mid - point property). By the Side - Side - Side (SSS) congruence criterion, which states that if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent, we can conclude that \(\triangle JIH\cong\triangle JKG\).

Answer:

1. Reason: Definition of a mid - point.
2. Reason: Side - Side - Side (SSS) congruence criterion.