If we want to prove that $△ A C D ≅ △ C A B,$ the first thing we need to do is to prove that $\overline{A B} ≅ \overline{D C} .$ We can also choose another path as there are multiple ways to solve this exercise.

First, let's prove that

△ A E B ≅ △ D E C .

Notice that

∠ A E B

and

∠ D E C

are vertical angles. Using the Vertical Angles Theorem, we can make the following conclusion.

∠ A E B ≅ ∠ D E C

Therefore, we can use the SAS Congruence Theorem. to prove that

△ A E B ≅ △ D E C .

Since the corresponding parts of the congruent triangles are congruent, we can also conclude the following.

\overline{A B} ≅ \overline{D C}

Let's redraw the figure using the conclusions that we have made.

As we can notice, $△ A C D$ and $△ C A B$ are right triangles and the hypotenuses and one of their legs are congruent. In this case, we use the Hypotenuse-Leg (HL) Theorem.

Hypotenuse-Leg (HL) Theorem
If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.

The illustration of the theorem is given below.

If △ P Q R and △ X Y Z right triangles, \overline{P R} ≅ \overline{X Z}, and \overline{P Q} ≅ \overline{X Y}, then △ P Q R ≅ △ X Y Z .

According to the theorem, we can immediately say that

△ A C D ≅ △ C A B .

Exercise