Question

Explain why ΔABC cannot be shown to be congruent to ΔCDA.

Answer 1

There isn't enough info to prove the triangles to be congruent or not. So we can't say for sure either way.

We have angle CAD = angle ACB given by the arc markings, and we know that AC = AC due to the reflexive theorem. However we are missing one third piece of information.

That third piece of info could be....

• AD = BC which allows us to use SAS
• angle ACD = angle CAB which allows us to use ASA
• angle ABC = angle CDA which allows us to use AAS (slight variation of ASA)

Since we don't know any of those three facts, we simply don't have enough information.

side note: If AB = CD, then this leads to SSA which is not a valid congruence theorem. If we had two congruent sides, the angle must be between the two sides, which is what AD = BC allows.