Chapter 3.3, Problem 13PSB

To determine

To prove : The congruency of $\overline{CB}$ with respect to $\overline{DB}$

Explanation of Solution

Given information : In this question we have given that $\angle A\cong \angle E,\overline{AB}\cong \overline{BE},\overline{FB}\perp \overline{AB}$ and $\angle 2\cong \angle 3$

Formula used: ASA Congruency rule.

Proof: Here, it is given that $\overline{FB}\perp \overline{AB}$ ,

  $\Rightarrow$ $\angle FBA\cong \angle FBE\cong 90°$

  $\Rightarrow \angle 1+\angle 2\cong 90°$ and $\angle 3+\angle 4\cong 90°$

  $\Rightarrow \angle 1\cong 90°-\angle 2$ and $\angle 4\cong 90°-\angle 3$

Now, as it is given that $\angle 2\cong \angle 3$ ,

  $\Rightarrow \angle 1\cong 90°-\angle 2$ and $\angle 4\cong 90°-\angle 2$

  $\Rightarrow \angle 1\cong \angle 4$

Now, In $\Delta ABCand\Delta EBD$ ,

  $\angle A\cong \angle E$ (Given)

  $\overline{AB}\cong \overline{BE}$ (Given)

  $\angle 1\cong \angle 4$ (Proved above)

  $\therefore \Delta ABC\cong \Delta EBD$ (BY ASA Congruency rule)

  $\Rightarrow \overline{CB}\cong \overline{DB}$ (BY CPCT)