Answer: Choice B
AAS and CPCTC
===============================================
Step-by-step explanation:
We're told that angle A = angle C. We have one pair of congruent angles.
The other pair of congruent angles we know about is angle ABD = angle CBD. This is because angle ABC is bisected by BD. The angle has been split into two equal pieces.
Since we have two pairs of congruent angles, we'll use either of the following
They are slightly different theorems. The order is important. With AAS, the side is not between the angles. With ASA, the side is between the angles.
The last piece of information we need is BD = BD. This is true because of the reflexive theorem. Notice how BD is not between the angles mentioned previously.
Since BD is not between those pairs of angles, we will go with AAS.
The triangles are congruent by AAS. See the diagram below.
Then we use CPCTC to conclude that AB = CB.
--------------
Notes:
- HL = hypotenuse leg
- AAS = angle angle side
- ASA = angle side angle
- SAS = side angle side
- CPCTC = corresponding parts of congruent triangles are congruent. This is a long acronym that I tend to think of as "CPC TC" so it's easier to remember. Feel free to use whatever other method you prefer.
- The HL theorem can only be applied to right triangles. It's not clear if the triangles are right triangles or not, so we cannot use HL here. Also, HL cannot be used since we're trying to prove that AB = CB in the first place.
- We can't use SAS since we don't have info about two pairs of congruent sides.