Question

∠gjk ≅ ∠hij, gk ⊥ jk, ij ≅ jk, and hi ⊥ ij. complete the proof that △gjk ≅ △hij.

Explanation:

Step1: Determine the right - angle

Since \(GK\perp JK\) and \(HI\perp IJ\), \(\angle GKJ=\angle HIJ = 90^{\circ}\) (Definition of perpendicular lines).

Step2: Use the Angle - Side - Angle (ASA) criterion

We know that \(\angle GJK=\angle HIJ\) (given), \(JK = IJ\) (given), and \(\angle GKJ=\angle HIJ\) (from Step 1).
By the ASA (Angle - Side - Angle) congruence criterion, if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the two triangles are congruent.
In \(\triangle GJK\) and \(\triangle HIJ\), \(\angle GJK=\angle HIJ\), \(JK = IJ\), \(\angle GKJ=\angle HIJ\)
So, \(\triangle GJK\cong\triangle HIJ\)

Answer:

\(\triangle GJK\cong\triangle HIJ\) (by ASA congruence criterion)