91Ó°ÊÓ

Angle bisector divides an angle into two equal parts.

From the figure in step 1 it can be observed that$\stackrel{¯}{YO}$ bisects$∠XYZ$ and$\stackrel{¯}{ZO}$ bisects $∠XZY$.

Therefore,$∠1â‰\dots ∠2$ and $∠3â‰\dots ∠4$, that is, $m∠1=m∠2\text{and}m∠3=m∠4$.

If the two sides of a triangle are congruent, then the angles opposite to those sides are congruent.

It is given that,$\stackrel{¯}{XY}â‰\dots \stackrel{¯}{XZ}$ therefore by isosceles triangle theorem $∠XYZâ‰\dots ∠XZY$.

But$m∠XYZ=m∠1+m∠2$ and $m∠XZY=m∠3+m∠4$.

Then $m∠1+m∠2=m∠3+m∠4$.

Substitute $m∠2$for $m∠1$and $m∠3$for $m∠4$into $m∠1+m∠2=m∠3+m∠4$and simplify.

$\begin{array}{c}m∠2+m∠2=m∠3+m∠3\\ 2m∠2=2m∠3\\ m∠2=m∠3\end{array}$

As$∠2â‰\dots ∠3$,therefore by converse isosceles triangle theorem, $\stackrel{¯}{YO}â‰\dots \stackrel{¯}{ZO}$.

Hence it is proved that $\stackrel{¯}{YO}â‰\dots \stackrel{¯}{ZO}$.

Write a two-column proof based on above explanation.