Question

analyzing diagrams
can you conclude that triangle ghf is congruent to triangle gjk? explain.
yes, by asa
yes, by aas
yes, by aaa
not enough information given

Explanation:

Step1: Identify Given Information

We have two right triangles \( \triangle GHF \) and \( \triangle GJK \). \( \angle H \) and \( \angle J \) are right angles (so \( \angle H = \angle J = 90^\circ \)). The segments \( GH \) and \( GJ \) are marked as equal (so \( GH = GJ \)). The angle \( \angle HGF \) and \( \angle JGK \) are vertical angles? Wait, no, actually, let's check the ASA criterion. ASA (Angle - Side - Angle) requires two angles and the included side to be equal. Here, \( \angle H = \angle J = 90^\circ \), \( GH = GJ \) (marked), and \( \angle HGF \) and \( \angle JGK \): Wait, actually, the angle at \( G \): Let's see, the triangles share the vertical angles? Wait, no, \( \angle HGF \) and \( \angle JGK \) are vertical angles? Wait, no, \( \angle HGF \) and \( \angle JGK \): Wait, the line \( HJ \) with \( G \) in the middle, \( GH = GJ \), right angles at \( H \) and \( J \), and \( \angle HGF \) and \( \angle JGK \): Wait, actually, the angle at \( G \) for both triangles: \( \angle HGF \) and \( \angle JGK \) are vertical angles? No, wait, \( \angle HGF \) and \( \angle JGK \): Wait, no, let's re - examine. The triangles are \( \triangle GHF \) and \( \triangle GJK \). We have \( \angle H = \angle J = 90^\circ \), \( GH = GJ \) (marked), and \( \angle HGF=\angle JGK \) (vertical angles are equal). So that's two angles ( \( \angle H \) and \( \angle HGF \)) and the included side \( GH \) for \( \triangle GHF \), and two angles ( \( \angle J \) and \( \angle JGK \)) and the included side \( GJ \) for \( \triangle GJK \). Since \( \angle H=\angle J \), \( GH = GJ \), and \( \angle HGF=\angle JGK \), this satisfies ASA (Angle - Side - Angle) congruence criterion.

Step2: Evaluate Other Options

• AAS (Angle - Angle - Side) requires two angles and a non - included side. But here we have the included side ( \( GH \) and \( GJ \)) between the two angles, so ASA is more appropriate.
• AAA (Angle - Angle - Angle) only gives similarity, not congruence.
• "Not enough information" is wrong because we have two angles (right angle and the vertical angle) and the included side equal.

Answer:

A. yes, by ASA