Question

question 9 of 10
assume that \\( \triangle a b c \cong \triangle p q r \\). which of the following congruence statements are correct? check all that apply.
a. \\( \overline{b c} \cong \overline{q r} \\)
b. \\( \angle r \cong \angle c \\)
c. \\( \overline{a c} \cong \overline{p q} \\)
d. \\( \angle b \cong \angle q \\)
e. \\( \angle a \cong \angle q \\)
f. \\( \overline{b c} \cong \overline{p q} \\)

Explanation:

Step1: Analyze congruent triangles' corresponding parts

When \(\triangle ABC\cong\triangle PQR\), the corresponding vertices are \(A\) with \(P\), \(B\) with \(Q\), and \(C\) with \(R\). Corresponding sides and angles are congruent.

• For side \(BC\), its corresponding side is \(QR\) (since \(B\) corresponds to \(Q\) and \(C\) corresponds to \(R\)). So \(\overline{BC}\cong\overline{QR}\) (matches option A).
• For angle \(R\), its corresponding angle is \(C\) (as \(C\) and \(R\) are corresponding vertices). So \(\angle R\cong\angle C\) (matches option B).
• For angle \(B\), its corresponding angle is \(Q\) (because \(B\) and \(Q\) are corresponding vertices). So \(\angle B\cong\angle Q\) (matches option D).

Step2: Check incorrect options

• For option C: \(\overline{AC}\) should correspond to \(\overline{PR}\), not \(\overline{PQ}\).
• For option E: \(\angle A\) should correspond to \(\angle P\), not \(\angle Q\).
• For option F: \(\overline{BC}\) corresponds to \(\overline{QR}\), not \(\overline{PQ}\).

Answer:

A. \(\overline{BC}\cong\overline{QR}\)
B. \(\angle R\cong\angle C\)
D. \(\angle B\cong\angle Q\)