Chapter 4.4, Problem 18WE

To determine

To prove: $\overline{YO}\cong \overline{ZO}$ using two column method.

Explanation of Solution

Given:

The given figure is,

  

And the condition is,

  $\overline{XY}\cong \overline{XZ}$ , $\overrightarrow{YO}$ bisects $\angle XYZ$ and $\overrightarrow{ZO}$ bisects $\angle XZY$

Statement :	Reason:
$\overline{XY}\cong \overline{XZ}$	Given
$m\angle 1=m\angle 2$	$\overrightarrow{YO}$ bisects $\angle XYZ$
$m\angle 3=m\angle 4$	$\overrightarrow{ZO}$ bisects $\angle XZY$
$m\angle 1+m\angle 2=m\angle 3+m\angle 4$	Isosceles triangle property (for $\Delta XYZ$)
$m\angle 2+m\angle 2=m\angle 3+m\angle 3$	Since $m\angle 1=m\angle 2$ and $m\angle 3=m\angle 4$
$\begin{array}{l}2m\angle 2=2m\angle 3\\ \Rightarrow m\angle 2=m\angle 3\end{array}$	Simplification
$\overline{YO}\cong \overline{ZO}$	Isosceles triangle property (for $\Delta OYZ$)
$\overline{YO}\cong \overline{ZO}$	Proved