Question

4
according to the aas corollary, which of the following sets of relationships can be used to show that δdac ≅ δbca?
a. ∠dca ≅ ∠bac and dc ≅ ab
b. ∠adc ≅ ∠cba and ∠bac ≅ ∠dca
c. ∠acb ≅ ∠cda and ∠bac ≅ ∠dca
d. ∠adc ≅ ∠cba and ad ≅ bc

Explanation:

Step1: Recall AAS Congruence

AAS (Angle - Angle - Side) congruence states that if two angles and a non - included side of one triangle are congruent to the corresponding two angles and non - included side of another triangle, the triangles are congruent.

Step2: Analyze Triangle \( \triangle DAC \) and \( \triangle BCA \)

We need to find the relationships that satisfy AAS for \( \triangle DAC\cong\triangle BCA \).

• For option A: \( \angle DCA\cong\angle BAC \) and \( DC\cong AB \). Let's check the triangles. In \( \triangle DAC \) and \( \triangle BCA \), this is not a valid AAS (or other congruence) setup as the side and angle are not corresponding correctly.
• For option B: \( \angle ADC\cong\angle CBA \) and \( \angle BAC\cong\angle DCA \). We also have the common side \( AC = CA \). In \( \triangle DAC \) and \( \triangle BCA \), \( \angle ADC\cong\angle CBA \), \( \angle BAC\cong\angle DCA \) and \( AC \) is a non - included side (for \( \angle ADC,\angle DCA \) in \( \triangle DAC \) and \( \angle CBA,\angle BAC \) in \( \triangle BCA \)). This satisfies AAS.
• For option C: \( \angle ACB\cong\angle CDA \) and \( \angle BAC\cong\angle DCA \). The angles and sides do not correspond correctly to satisfy AAS for \( \triangle DAC\cong\triangle BCA \).
• For option D: \( \angle ADC\cong\angle CBA \) and \( AD\cong BC \). The side and angle do not correspond in a way that satisfies AAS for the two triangles.

Answer:

B. \( \angle ADC\cong\angle CBA \) and \( \angle BAC\cong\angle DCA \)