Question

Write proofs in two-column form.

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

Prove:$\mathbf{m}\angle \mathbf{1}\cong \mathbf{m}\angle \mathbf{4}$

Short answer

Statement	Reason
$\overline{\mathrm{XY}}\cong \overline{\mathrm{XZ}}$	Given
$\mathbf{m}\angle \mathbf{1}\mathbf{+}\mathbf{m}\angle \mathbf{2}\mathbf{=}\mathbf{m}\angle \mathbf{3}\mathbf{+}\mathbf{m}\angle \mathbf{4}$	Isosceles triangle theorem
$\overline{\mathrm{OY}}\cong \overline{\mathrm{OZ}}$	Given
$\mathbf{m}\angle \mathbf{2}\mathbf{=}\mathbf{m}\angle \mathbf{3}$	Isosceles triangle theorem
$\mathbf{m}\angle \mathbf{1}\mathbf{=}\mathbf{m}\angle \mathbf{4}$	Since , so cancels out
$\mathbf{m}\angle \mathbf{1}\mathbf{=}\mathbf{m}\angle \mathbf{4}$	Proved

Step-by-step solution

Step 1. 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 2. Apply isosceles triangle theorem to ΔOYZ.

It is given that, $\overline{OY}\cong \overline{OZ}$, therefore by isosceles triangle theorem, $\angle 2\cong \angle 3$, that is $m\angle 2=m\angle 3$.

Step 3. Description of step.

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

$\begin{array}{c}m\angle 1+m\angle 2=m\angle 2+m\angle 4\\ m\angle 1=m\angle 4\end{array}$

Hence it is proved that $m\angle 1=m\angle 4$.

Step 4. Two column proof.

Write a two-column proof based on above explanation.

Statement	Reason
$\overline{XY}\cong \overline{XZ}$	Given
$m\angle 1+m\angle 2=m\angle 3+m\angle 4$	Isosceles triangle theorem
$\overline{OY}\cong \overline{OZ}$	Given
$m\angle 2=m\angle 3$	Isosceles triangle theorem
$m\angle 1=m\angle 4$	Since , so cancels out
$m\angle 1=m\angle 4$	Proved