Question
Given: $\overline{B A} \cong \overline{D C}, \overline{B A} \perp \overline{A D}$ and $\overline{A D} \perp \overline{D C}$.
Prove: $\triangle B A D \cong \triangle C D A$.
Step
1
try
Statement
\[
\overline{B A} \cong \overline{D C}
\]
\[
\begin{array}{l}
\overline{B A} \perp \overline{A D} \\
\overline{A D} \perp \overline{D C}
\end{array}
\]

Reason

Given

Type of Statement

Step 1 ：\(\overline{BA} \cong \overline{DC}\) (Given)

Step 2 ：\(\overline{BA} \perp \overline{AD}\) and \(\overline{AD} \perp \overline{DC}\) (Given)

Step 3 ：\(\angle BAD = \angle CDA\) (Both are right angles, so they are congruent)

Step 4 ：\(\overline{AD} \cong \overline{AD}\) (Reflexive property of congruence)

Step 5 ：\(\triangle BAD \cong \triangle CDA\) (By the Hypotenuse-Leg (HL) Theorem, if a right triangle has a hypotenuse and a leg that are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent)

Step 6 ：Check: The triangles \(\triangle BAD\) and \(\triangle CDA\) are congruent, which meets the requirements of the problem