Chapter 12, Problem 3CR

To determine

To prove : The given quadrilateral $ABCD$ is a parallelogram.

Explanation of Solution

Given information :

  $\begin{array}{l}\overline{BD}\perp \overline{AD}\dots \left(i\right)\\ \overline{BD}\perp \overline{BC}\dots \left(ii\right)\\ \overline{AB}\cong \overline{CD}\dots \left(iii\right)\end{array}$

Formula used :

Opposite sides of a quadrilateral ABCD are parallel.

Proof :

From $\left(i\right)\&\left(ii\right)$ , $\begin{array}{c}\overline{BD}\perp \overline{AD}\\ \overline{BD}\perp \overline{BC}\end{array}\}\Rightarrow$

  $\overline{AD}\cong \overline{BC}\dots \left(iv\right)$

  

From $\left(iii\right)\text{ \& }\left(iv\right)$ ,

  $\overline{AB}\text{ parallel to }\overline{CD}$ and $\overline{AD}\text{ parallel to }\overline{BC}$

Therefore, the given quadrilateral $ABCD$ is a parallelogram.