Question

practice with congruency theorems and cpctc.
which congruency theorem can be used to prove that $\triangle abd \cong \triangle dca$?
\bigcirc sss
\bigcirc asa (marked with red cross)
\bigcirc sas
\bigcirc aas

Explanation:

Step1: Analyze given information

From the diagram, \( AB = DC \) (marked with one tick), \( \angle BAD=\angle CDA = 90^\circ \) (right angles), and \( AD = DA \) (common side).

Step2: Identify congruence theorem

We have two sides and the included angle? Wait, no: \( AB = DC \), \( \angle BAD=\angle CDA \), \( AD = DA \). So two sides and the included angle? Wait, \( \angle BAD \) is between \( AB \) and \( AD \), \( \angle CDA \) is between \( DC \) and \( DA \). So \( AB = DC \), \( \angle BAD=\angle CDA \), \( AD = DA \). So this is SAS (Side - Angle - Side) because we have two sides and the included angle equal. Wait, let's check again:

• \( AB = DC \) (given by the tick marks)
• \( \angle BAD=\angle CDA = 90^\circ \) (right angles)
• \( AD = DA \) (common side, reflexive property)

So the sides \( AB \) and \( AD \) with included angle \( \angle BAD \) in \( \triangle ABD \), and sides \( DC \) and \( DA \) with included angle \( \angle CDA \) in \( \triangle DCA \). So by SAS congruence theorem, \( \triangle ABD\cong\triangle DCA \).

Wait, but let's check the options. The options are SSS, ASA, SAS, AAS. We have two sides (AB = DC, AD = DA) and the included angle ( \( \angle BAD=\angle CDA \)) equal. So SAS is the correct theorem.

Wait, the original marked answer was ASA, which is wrong. Let's re - evaluate:

ASA requires two angles and the included side. Let's see the angles: \( \angle BAD=\angle CDA = 90^\circ \), but what about the other angles? We don't have information about another pair of angles and the included side. But for SAS: we have \( AB = DC \), \( \angle BAD=\angle CDA \), \( AD = DA \). So SAS is correct.

Answer:

The correct congruency theorem is SAS. So the answer is the option with "SAS" (e.g., if the options are labeled as: \(\bigcirc\) SSS, \(\bigcirc\) ASA, \(\bigcirc\) SAS, \(\bigcirc\) AAS, then the answer is the option with SAS, for example, if it's the third option: "SAS" (the option with the circle next to SAS).