Question

triangle ghj is rotated 90° about point x, resulting in triangle str. which congruency statement is true?
○ \\( \overline { t r } \cong \overline { g j } \\)
○ \\( \angle s \cong \angle h \\)
○ \\( \overline { t s } \cong \overline { h g } \\)
○ \\( \angle r \cong \angle g \\)

Explanation:

Step1: Recall the properties of rotation

Rotation is a rigid transformation. So, \(\triangle GHJ\cong\triangle STR\).

Step2: Use the congruence of triangles

If \(\triangle GHJ\cong\triangle STR\), then corresponding parts are congruent.

• For \(\overline{TS}\) and \(\overline{HG}\):

Since \(\triangle GHJ\cong\triangle STR\), by the order of congruence ( \(G - H - J\) and \(S - T - R\) ), \(\overline{HG}\) corresponds to \(\overline{TS}\) (because in congruent triangles, the order of vertices matters for corresponding sides and angles. If \(\triangle GHJ\cong\triangle STR\), then \(HG\) in \(\triangle GHJ\) and \(TS\) in \(\triangle STR\) are corresponding sides). So, \(\overline{TS}\cong\overline{HG}\) by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem.

• For \(\overline{TR}\) and \(\overline{GJ}\):

\(\overline{TR}\) corresponds to \(\overline{HJ}\) (not \(\overline{GJ}\)), so \(\overline{TR}
ot\cong\overline{GJ}\).

• For \(\angle S\) and \(\angle H\):

\(\angle S\) corresponds to \(\angle G\) (not \(\angle H\)), so \(\angle S
ot\cong\angle H\).

• For \(\angle R\) and \(\angle G\):

\(\angle R\) corresponds to \(\angle J\) (not \(\angle G\)), so \(\angle R
ot\cong\angle G\).

Answer:

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