Question

△ghj is equilateral, ∠ihj ≅ ∠jgk, and (overline{gk} cong overline{hi}). complete the proof that (∠i ≅ ∠k).

Explanation:

Step1: Use the property of equilateral triangle

Since \(\triangle GHJ\) is equilateral, by the definition of an equilateral triangle, all its sides are equal. So, \(GJ = HJ\).

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

We know that \(GK\cong HJ\) (given), \(GJ = HJ\) (from step 1), so \(GK\cong GJ\). Also, \(\angle IHJ\cong\angle JGK\) (given).
In \(\triangle IHJ\) and \(\triangle KJG\):

• \(HJ = GJ\) (from step 1)
• \(\angle IHJ=\angle JGK\) (given)
• \(GK = HJ\) (given)

By the SAS (Side - Angle - Side) congruence criterion, \(\triangle IHJ\cong\triangle KJG\).

Step3: Use the property of congruent triangles

If two triangles \(\triangle IHJ\) and \(\triangle KJG\) are congruent, then their corresponding angles are equal. So, \(\angle I\cong\angle K\) (corresponding parts of congruent triangles are congruent).

Answer:

\(\angle I\cong\angle K\)