Question

1. \\( \triangle a b c \\) was transformed to create \\( \triangle p q r \\) using the transformation function

\\( (x, y) \
ightarrow(-x, y) \\) as shown.

which congruency statements accurately describe the corresponding sides and
angles of \\( \triangle a b c \\) and the image \\( \triangle p q r \\) ?

i	\\( \overline{a b} \cong \overline{p q} \\)
ii	\\( \angle b a c \cong \angle q r p \\)
iii	\\( \overline{a c} \cong \overline{p r} \\)
iv	\\( \angle c b a \cong \angle r q p \\)

Explanation:

Step1: Analyze the transformation

The transformation \((x,y)\to(-x,y)\) is a reflection over the \(y -\)axis. A reflection is a rigid transformation, so \(\triangle ABC\cong\triangle PQR\).

Step2: Determine corresponding parts

For \(\triangle ABC\) and \(\triangle PQR\), \(A\) corresponds to \(R\), \(B\) corresponds to \(Q\), and \(C\) corresponds to \(P\).

• For side \(\overline{AB}\) and \(\overline{RQ}\), since \(\triangle ABC\cong\triangle PQR\), \(\overline{AB}\cong\overline{RQ}\) (not \(\overline{PQ}\), so statement I is false).
• \(\angle BAC\) corresponds to \(\angle QRP\) (because \(A\to R\) and \(C\to P\)), so \(\angle BAC\cong\angle QRP\) (statement II is true).
• \(\overline{AC}\) corresponds to \(\overline{RP}\), so \(\overline{AC}\cong\overline{PR}\) (statement III is true).
• \(\angle CBA\) corresponds to \(\angle RQP\) (because \(B\to Q\) and \(A\to R\), \(C\to P\)), so \(\angle CBA\cong\angle RQP\) (statement IV is true).

Answer:

II, III, and IV