Subjects geometry

Congruence Similarity

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Congruence Similarity


1. The problem states that triangle ABC is congruent to triangle PQR and that \( \angle BAC = \angle QPR \). We are asked to find which side in triangle PQR is equal to side BC in triangle ABC. 2. Since triangles are congruent, all corresponding sides and angles are equal. Given the options, side BC corresponds to side QR in triangle PQR. 3. Therefore, \( BC = QR \). 4. Next, we need to show that \( \triangle PQR \) and \( \triangle TUV \) are congruent. 5. From the description, both triangles have a right angle (at \( Q \) in \( \triangle PQR \) and at \( U \) in \( \triangle TUV \)). 6. Side \( PQ \) in \( \triangle PQR \) has the same marking as \( TU \) in \( \triangle TUV \), indicating \( PQ = TU \). 7. Side \( QR \) in \( \triangle PQR \) has the same marking as \( UV \) in \( \triangle TUV \), indicating \( QR = UV \). 8. Therefore, \( \triangle PQR \) and \( \triangle TUV \) have two corresponding sides equal and the included right angle equal. By the SAS (Side-Angle-Side) congruency criterion, \( \triangle PQR \cong \triangle TUV \). Final answers: - Side equal to \( BC \) in \( \triangle PQR \) is \( QR \). - \( \triangle PQR \cong \triangle TUV \) by SAS congruency.