Chapter 6, Problem 20CR

To determine

To prove : The congruency of $\overline{BE}$ with respect to $\overline{CD}$ .

Explanation of Solution

Given information : Here, we have given that $\overline{PA}\perp m$ and $\overline{PD}\cong \overline{PE}$

Proof: In $\Delta PAD$ and $\Delta PAE$

  $\overline{PD}\cong \overline{PE}$ (Given)

  $\overline{PA}\cong \overline{PA}$ (Common)

  $\angle DPA\cong \angle EPA$ (as, $\overline{PA}\perp m$ )

  $\Rightarrow \Delta PAD\cong \Delta PAE$ (By SAS Congruency Rule)

  $\therefore \overline{AD}\cong \overline{AE}$ (By Corresponding Part of Congruent Triangles)

Now, In $\Delta ABE$ and $\Delta ACD$ ,

  $\overline{AE}\cong \overline{AD}$ (Proved above)

  $\angle A\cong \angle A$ (Common)

  $\overline{AB}\cong \overline{AC}$ (Radius of the circle)

  $\Rightarrow \Delta ABE\cong \Delta ACD$ (By SAS Congruency Rule)

  $\therefore \overline{BE}\cong \overline{CD}$ (By Corresponding Part of Congruent Triangles)