Question

1. which congruency statement would not result in △bcd ≅ △qrs? (1 point)

a bc ≅ qr forms the side - angle - side congruence theorem
b ∠d ≅ ∠s forms the angle - side - angle congruence theorem
c cd ≅ rs forms the side - angle - side congruence theorem
d bc ≅ qr and cd ≅ rs forms the side - side - side congruence theorem

Explanation:

Step1: Recall the SAS (Side - Angle - Side) Congruence Theorem

For two triangles \(\triangle BCD\) and \(\triangle QRS\), if two sides and the included angle of one triangle are congruent to two sides and the included angle of the other triangle, the triangles are congruent.

Step2: Analyze Option A

If \(\overline{BC}\cong\overline{QR}\) (side), \(\angle B\cong\angle Q\) (angle, as they are the included angles in the triangles), and \(\overline{CD}\cong\overline{RS}\) (side), then by SAS, \(\triangle BCD\cong\triangle QRS\).

Step3: Analyze Option B

If \(\angle D\cong\angle S\), in \(\triangle BCD\) and \(\triangle QRS\), we do not know if these angles are included angles between the sides. For example, if we consider the sides \(\overline{BC}\cong\overline{QR}\) and \(\overline{CD}\cong\overline{RS}\), the included angles for SAS are \(\angle B\) and \(\angle Q\) (not \(\angle D\) and \(\angle S\)). So, \(\angle D\cong\angle S\) does not form the Angle - Side - Angle (ASA) or Side - Angle - Side (SAS) congruence.

Step4: Analyze Option C

If \(\overline{CD}\cong\overline{RS}\), along with \(\overline{BC}\cong\overline{QR}\) (sides) and \(\angle B\cong\angle Q\) (included angle), by SAS, \(\triangle BCD\cong\triangle QRS\).

Step5: Analyze Option D

If \(\overline{BC}\cong\overline{QR}\) and \(\overline{CD}\cong\overline{RS}\), and \(\angle B\cong\angle Q\) (included angle), by SAS, \(\triangle BCD\cong\triangle QRS\).

Answer:

B. \(\angle D\cong\angle S\) forms the Angle - Side - Angle Congruence Theorem