Chapter 3, Problem 24CR

To determine

To calculate: The congruency of $\overline{ST}$ with respect to $\overline{RV}$

Explanation of Solution

Given information:$\angle NOT\cong \angle POV\text{, }\angle N\cong \angle P$ and $O$ is a midpoint

Formula used:ASA congruency rule and SAS congruency rules.

Proof:

In $\Delta NOT$ and $\Delta POV$

  $\angle NOT=\angle POV$ (Given)

  $\angle N=\angle P$ (Given)

  $NO=PO$ (O is midpoint)

  $\therefore \Delta NOT\cong \Delta POV$ (By angel side angel congruency rule)

  $\therefore \overline{TO}=\overline{VO}$ (Corresponding parts of congruent triangles are equal) $\mathrm{...}(1)$

Now, in $\Delta RON$ and $\Delta SOP$

  $NO=PO$ (O is midpoint)

  $\angle N=\angle P$ (Given)

  $\angle NOT=\angle POV$ (Given)

  $\angle NOT+\angle VOT=\angle POV+\angle VOT$ (Adding $\angle VOT$ both sides)

  $\angle VON=\angle TOP$

Or $\angle RON=\angle SOP$

  $\therefore \overline{RN}=\overline{SP}$ (Because lines opposite to equal angels are equal)

  $\therefore \Delta RON\cong \Delta SOP$ (By angel side angel congruency rule)

  $\therefore \overline{RO}=\overline{SO}$ (Corresponding parts of congruent triangles are equal) $\mathrm{...}(2)$

From 1 and 2

  →$\overline{RO}=\overline{SO}$

  $\overline{RV}+\overline{VO}=\overline{ST}+\overline{TO}$

  $\therefore \overline{RV}=\overline{ST}$ (Because $\overline{VO}=\overline{TO}$ )

  $$