If triangle ACD is similar to triangle ABE, find the value of x

A. "Segment AD bisects angle CAB." is the right answer.

Step-by-step explanation:

Given : In ΔABC ,AC≅AB.

⇒∠ACB=∠CBA(1) (∵ angles opposite to equal sides of a triangle are equal )

Now in ΔACD and ΔABD

AD=AD (common)(2)

Here we need one more statement to prove the triangles congruent that is only statement (A) fits in it.

If AD bisects ∠CAB then ∠CAD=∠BAD..(3)

Now again Now in ΔACD and ΔABD

∠ACB=∠CBA [from (1)]

AD=AD [common]

∠CAD=∠BAD [from (3)]

So by ASA congruency criteria ΔADC≅ΔABD.