Question

In the figure below, segment AC is congruent to segment AB:

Which statement is used to prove that angle ABD is congruent to angle ACD?

A. Segment AD bisects angle CAB.
B. Triangle ACD is similar to triangle ABD.
C. Segment AD is congruent to segment AC.
D. Angle CAB is congruent to angle CBA.

Answer 1

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

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.