Question

triangle ghj is rotated 90° about point x, resulting in triangle str. which congruency statement is true?
o tr ≅ gj
o ∠s ≅ ∠h
o ts ≅ hg
o ∠r ≅ ∠g

Explanation:

Step1: Recall the property of rotation

Rotation is a rigid transformation. Rigid transformations (rotations, translations, reflections) preserve side - lengths and angle - measures. So, if \(\triangle GHJ\) is rotated to get \(\triangle STR\), then \(\triangle GHJ\cong\triangle STR\).

Step2: Analyze the correspondence of vertices

When \(\triangle GHJ\) is rotated about point \(X\) to get \(\triangle STR\), the correspondence of vertices is \(G\to S\), \(H\to T\), \(J\to R\).

Step3: Check each option

• For the side - length:
• If \(G\to S\), \(H\to T\), \(J\to R\), then \(\overline{HG}\) corresponds to \(\overline{TS}\) (by the order of vertices in congruent triangles \(\triangle GHJ\cong\triangle STR\), \(HG\) in \(\triangle GHJ\) and \(TS\) in \(\triangle STR\)). Since congruent triangles have congruent corresponding parts, \(\overline{TS}\cong\overline{HG}\).
• \(\overline{TR}\) corresponds to \(\overline{HJ}\), not \(\overline{GJ}\).
• For the angle:
• \(\angle S\) corresponds to \(\angle G\), not \(\angle H\).
• \(\angle R\) corresponds to \(\angle J\), not \(\angle G\).

Answer:

\(\overline{TS}\cong\overline{HG}\) (i.e., the option \(\overline{TS}\cong\overline{HG}\) is correct)