The correct option is C. The two triangles are congruent by SAS property.
The image shows two triangles, ΔABD and ΔCDA, within a quadrilateral, and asks which congruency theorem can be used to prove that ΔABD is congruent to ΔCDA. The options are SSS, ASA, SAS, and AAS.
To solve this, let's consider the congruence theorems:
- SSS (Side-Side-Side): Three pairs of corresponding sides are equal.
- ASA (Angle-Side-Angle): Two pairs of corresponding angles and the side between them are equal.
- SAS (Side-Angle-Side): Two pairs of corresponding sides and the angle between them are equal.
- AAS (Angle-Angle-Side): Two pairs of corresponding angles and a non-included side are equal.
We need to look for these patterns in the triangles ΔABD and ΔCDA.
From the image, it seems that:
1. Line segment AD is shared by both triangles, so AD = AD (reflexive property).
2. Similarly, angles ADC and BAD are marked as equal, again likely due to the vertical angle theorem.
2. It appears that sides AB and CD are marked to indicate they are equal, likely by the vertical angle theorem since they are vertical angles.
Given this information, we can say that two angles and the included side between them are equal, which corresponds to the SAS (Side-Angle-Side) congruence theorem.
Thus, the correct option to prove that ΔABD is congruent to ΔCDA is SAS.
The complete question with diagram is given below: