Question

State whether the given information be used to prove that two lines are parallel and also state which lines.

Given: $\overline{\text{MN}}\cong \overline{\text{MP}}$; $\angle \text{NMO}\cong \angle \text{PMO}$.

Prove:$\overleftrightarrow{\text{MO}}$is the $\perp$bisector of $\overline{\text{NP}}$.

Short answer

$\overleftrightarrow{MO}$ is the perpendicular bisector of $\overline{NP}$.

Step-by-step solution

Step 1.  Consider the diagram. 

Here, $\overline{MN}=\overline{MP}$ and $\angle NMO\cong \angle PMO$.

Step 2.  Show the calculation.

Consider the $\Delta MNO$ and $\Delta MPO$:

$\overline{MN}\cong \overline{MP}$

$\angle NMO\cong \angle PMO$

$\overline{MO}\cong \overline{MO}$

So, by SAS Postulate, $\Delta MNO\cong \Delta MPO$.

So, $\overline{NO}\cong \overline{PO}$.

Hence, by Theorem 4-6, $\overleftrightarrow{MO}$ is the perpendicular bisector of $\overline{NP}$.

Step 3.  State the conclusion.

Therefore, $\overleftrightarrow{MO}$ is the perpendicular bisector of $\overline{NP}$ (proved).