Chapter 6, Problem 16RP

To determine

To prove : The line segment $\overline{BC}$ will bisect the line segment $\overline{AD}$ .

Explanation of Solution

Given information : Here, we have given that $m\parallel n$ , $p\parallel n$ and $\overline{AD}$ bisects $\overline{BC}$ .

Construction: Join $\overline{AB}$ and $\overline{CD}$ .

Proof:

As, $\overline{AD}$ bisects $\overline{BC}$ ,

  $\Rightarrow \overline{BE}\cong \overline{CE}$

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

  $\overline{BE}\cong \overline{CE}$ (Proved above)

  $\angle ABE\cong \angle DCE$ (Alternate Interior Angles)

  $\angle AEB\cong \angle DEC$ (Vertically Opposite Angles)

  $\Rightarrow \Delta ABE\cong \Delta DEC$ (By ASA Congruency Rule)

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

  $\Rightarrow$$\overline{BC}$ bisects $\overline{AD}$ .