Question

triangle ghj is rotated 90° about point x, resulting in triangle str. which congruency statement is true?

• $overline{tr} cong overline{gj}$
• $angle s cong angle h$
• $overline{ts} cong overline{hg}$
• $angle r cong angle g$

Explanation:

Step1: Recall Rotation Congruence

A rotation is a rigid transformation, so the image (△STR) is congruent to the pre - image (△GHJ). Corresponding sides and angles are congruent. We need to find the correct correspondence between the vertices of △GHJ and △STR.
When △GHJ is rotated 90° about point X to get △STR, the correspondence of vertices should be G→S, H→T, J→R (by looking at the position of the triangles and the rotation).

Step2: Analyze Each Option

• Option 1: \(\overline{TR}\cong\overline{GJ}\). Let's check the correspondence. If J→R and T→H (wait, no, from the rotation, the correct correspondence for sides: \(\overline{GH}\) corresponds to \(\overline{ST}\), \(\overline{HJ}\) corresponds to \(\overline{TR}\), \(\overline{GJ}\) corresponds to \(\overline{SR}\). So \(\overline{TR}\) corresponds to \(\overline{HJ}\), not \(\overline{GJ}\). So this is false.
• Option 2: \(\angle S\cong\angle H\). Since G→S and H→T, \(\angle S\) corresponds to \(\angle G\), not \(\angle H\). So this is false.
• Option 3: \(\overline{TS}\cong\overline{HG}\). Since T corresponds to H and S corresponds to G, \(\overline{TS}\) is the side between T and S, and \(\overline{HG}\) is the side between H and G. So \(\overline{TS}\) (corresponding to \(\overline{HG}\)) are congruent. Let's verify the vertex correspondence: G→S, H→T, J→R. So \(\overline{HG}\) (from H to G) and \(\overline{TS}\) (from T to S) are corresponding sides. Since rotation preserves side lengths, \(\overline{TS}\cong\overline{HG}\).
• Option 4: \(\angle R\cong\angle G\). Since R corresponds to J and G corresponds to S, \(\angle R\) corresponds to \(\angle J\), not \(\angle G\). So this is false.

Answer:

\(\boldsymbol{\overline{TS}\cong\overline{HG}}\) (the option with \(\overline{TS}\cong\overline{HG}\))