Question

Write proofs in two–column form.

Given:$\overline{\mathrm{XY}}\cong \overline{\mathrm{XZ}}\mathbf{;}$

$\overline{\mathrm{YO}}$bisects$\angle \mathbf{XYZ}$

$\overline{\mathrm{ZO}}$bisects$\angle \mathbf{XZY}$

Prove: $\overline{\mathrm{YO}}\cong \overline{\mathrm{ZO}}$

Short answer

Statement	Reason
$\overline{\mathrm{XY}}\cong \overline{\mathrm{XZ}}$	Given
$\mathbf{m}\angle \mathbf{1}\mathbf{=}\mathbf{m}\angle \mathbf{2}$	bisects
$\mathbf{m}\angle \mathbf{3}\mathbf{=}\mathbf{m}\angle \mathbf{4}$	bisects
$\mathbf{m}\angle \mathbf{1}\mathbf{+}\mathbf{m}\angle \mathbf{2}\mathbf{=}\mathbf{m}\angle \mathbf{3}\mathbf{+}\mathbf{m}\angle \mathbf{4}$	Isosceles triangle theorem
$\mathbf{m}\angle \mathbf{2}\mathbf{=}\mathbf{m}\angle \mathbf{3}$	Since
$\overline{\mathrm{YO}}\cong \overline{\mathrm{ZO}}$	Converse isosceles triangle theorem

Step-by-step solution

Step 1. Apply angle bisector property.

Angle bisector divides an angle into two equal parts.

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

Therefore,$\angle 1\cong \angle 2$ and $\angle 3\cong \angle 4$, that is, $m\angle 1=m\angle 2\text{and}m\angle 3=m\angle 4$.

Step 2. Apply isosceles triangle theorem to ΔXYZ.

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

It is given that,$\overline{XY}\cong \overline{XZ}$ therefore by isosceles triangle theorem $\angle XYZ\cong \angle XZY$.

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

Then $m\angle 1+m\angle 2=m\angle 3+m\angle 4$.

Step 3. Description of step.

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

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

Step 4. Apply converse of isosceles triangle theorem to ΔOYZ.

As$\angle 2\cong \angle 3$,therefore by converse isosceles triangle theorem, $\overline{YO}\cong \overline{ZO}$.

Hence it is proved that $\overline{YO}\cong \overline{ZO}$.

Step 5. Two column proof.

Write a two-column proof based on above explanation.

Statement	Reason
$\overline{XY}\cong \overline{XZ}$	Given
$m\angle 1=m\angle 2$	bisects
$m\angle 3=m\angle 4$	bisects
$m\angle 1+m\angle 2=m\angle 3+m\angle 4$	Isosceles triangle theorem
$m\angle 2=m\angle 3$	Since
$\overline{YO}\cong \overline{ZO}$	Converse isosceles triangle theorem