Exercise 7 Page 268

If we want to prove that $△ACD\cong △CAB,$ the first thing we need to do is to prove that $\overline{AB}\cong \overline{DC}.$ We can also choose another path as there are multiple ways to solve this exercise.

First, let's prove that $△AEB\cong △DEC.$ Notice that $\angle AEB$ and $\angle DEC$ are vertical angles. Using the Vertical Angles Theorem, we can make the following conclusion. Therefore, we can use the SAS Congruence Theorem. to prove that $△AEB\cong △DEC.$ Since the corresponding parts of the congruent triangles are congruent, we can also conclude the following. Let's redraw the figure using the conclusions that we have made.

As we can notice, $△ACD$ and $△CAB$ 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.

According to the theorem, we can immediately say that $△ACD\cong △CAB.$