Question

1. (\triangle abc) was transformed to create (\triangle pqr) using the transformation function ((x,y)rightarrow(-x,y)) as shown. which congruency statements accurately describe the corresponding sides and angles of (\triangle abc) and the image (\triangle pqr)? (\begin{array}{|l|l|}hline i & overline{ab} cong overline{pq} \\ hline ii & angle bac cong angle qrp \\ hline iii & overline{ac} cong overline{pr} \\ hline iv & angle cba cong angle rqp \\ hlineend{array})

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 \(\triangle PQR\) (where \(A\to R\), \(B\to Q\), \(C\to P\) due to the reflection \((x,y)\to(-x,y)\)):

• For side - side congruence:
• \(\overline{AB}\) corresponds to \(\overline{RQ}\). But if we consider the order of vertices in congruence statements for triangles \(\triangle ABC\) and \(\triangle PQR\) (since \(\triangle ABC\cong\triangle PQR\) by reflection), \(\overline{AB}\cong\overline{PQ}\) (because of the nature of rigid transformation, lengths are preserved).
• \(\overline{AC}\) corresponds to \(\overline{RP}\), so \(\overline{AC}\cong\overline{PR}\) (lengths are preserved in a reflection).
• For angle - angle congruence:
• \(\angle BAC\) in \(\triangle ABC\) corresponds to \(\angle QRP\) in \(\triangle PQR\) (angles are preserved in a reflection).
• \(\angle CBA\) in \(\triangle ABC\) corresponds to \(\angle RQP\) in \(\triangle PQR\) (angles are preserved in a reflection).

Answer:

All statements are true.