Triangle Congruence with Parallel Lines: Proving CDA ≅ ABC

AB is parallel to CD.

Determine which of the following options needs to be true in order that the triangle CDA is equal to the triangle ABC:

In order to answer the question, we need to know all four congruence theorems -

SAS, SAA, ASA, SSA

Now let's deduce which data we can prove from the question -

AB is parallel to DC

And from this it follows that angle BAC equals angle ACD, given that these are equal alternate angles,

Furthermore we have a shared side AC, hence we have a shared angle and side.

We could have proven congruence using AB=DC, and then we would have SAA,

However this is not given to us in the options,

Hence let's look at the fourth congruence theorem - SSA,

In order for it to work, we need to show another side, BC=AD,

and this is indeed one of the options!

However it's important to remember that the fourth congruence theorem has a condition.

The theorem is valid only if the angle is opposite to the larger of the two sides.

Therefore we need to know that ACAC,

and indeed, one of these options exists!

Thus, we can see that there are two things we need,

and therefore answer D is correct!