Question

triangles pqr and uvw, shown below, are congruent because there exists a series of translations, rotations, and reflections called t that carries △pqr onto △uvw. which statement must be true? a. $overline{pq}congoverline{uv}$ because t followed by a reflection across $overline{pr}$ carries $overline{pq}$ onto $overline{uv}$. b. $overline{qr}congoverline{vw}$ because t carries $overline{qr}$ onto $overline{vw}$. c. $angle rcongangle u$ because t followed by a translation to the right carries $angle r$ onto $angle u$. d. $angle qcongangle w$ because t carries $angle q$ onto $angle w$.

Explanation:

Step1: Recall congruent - triangle property

If a transformation \(T\) maps \(\triangle PQR\) onto \(\triangle UVW\), corresponding parts of congruent triangles are congruent. When two triangles are congruent, corresponding sides and corresponding angles are equal.

Step2: Analyze option A

If \(T\) maps \(\triangle PQR\) onto \(\triangle UVW\), the corresponding side to \(\overline{PQ}\) is \(\overline{UV}\). Since the transformation \(T\) carries \(\triangle PQR\) onto \(\triangle UVW\), we have \(\overline{PQ}\cong\overline{UV}\).

Step3: Analyze option B

There is no information suggesting that \(T\) specifically carries \(\overline{QR}\) onto \(\overline{VW}\) in the way described. Just because the triangles are congruent doesn't mean this particular mapping of sides is described by the general transformation \(T\) as stated.

Step4: Analyze option C

There is no indication that \(T\) followed by a translation maps \(\angle R\) onto \(\angle U\) in the given description.

Step5: Analyze option D

There is no evidence that \(T\) carries \(\angle Q\) onto \(\angle W\) as described.

Answer:

A. \(\overline{PQ}\cong\overline{UV}\) because \(T\) followed by a reflection across \(\overline{PR}\) carries \(\overline{PQ}\) onto \(\overline{UV}\)