Chapter 12.5, Problem 13E

To determine

To proof: $△BAD\cong △CDA$

Explanation of Solution

Given information:

  $\begin{array}{l}\overline{AC}\cong \overline{BD}\\ \overline{AB}\perp \overline{AD}\\ \overline{CD}\perp \overline{AD}\end{array}$

  

Formula used:

The Hypotenuse-Leg Congruence Theorem

If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle , then the two triangles are congruent.

Proof:

According to the question,

We have two triangles :-

     

Triangles $△BAD\text{ and }△CDA.$

Now,

Statements	Reasons
H  1. $\overline{AC}=\overline{BD}$   2. $\overline{AB}\perp \overline{AD}$ , $\overline{CD}\perp \overline{AD}$   3. $\angle A\text{ and }\angle D\text{ are right angles}\text{.}$   4. $△BAD\text{ and }△CDA\text{ are right triangles}\text{.}$	1.Given 2.Given 3. Definition of $\perp$ lines. 4. Definition of a right triangle.
L  5. $\overline{AD}=\overline{DA}$   6. $△BAD\cong △CDA$	5. Reflexive Property of Congruence 6. HL Congruence Theorem

Hence Proved.