Question

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

Given: $\overline{\text{MO}}\perp \overline{\text{NP}}$; $\overline{\text{NO}}\cong \overline{\text{PO}}$.

Prove: $\overline{\text{MN}}\cong \overline{\text{MP}}$.

Short answer

It is proved $\overline{NM}\cong \overline{MP}$.

Step-by-step solution

Step 1.  Consider the diagram. 

Here, $\overline{MO}\perp \overline{NP}$ and $NO\cong PO$.

Step 2.  Show the calculation.

Consider the right angled $\Delta NQO$ and $\Delta PQO$:

$\overline{NO}\cong \overline{PO}$, which are hypotenuse.

$\overline{QO}\cong \overline{QO}$, which are legs.

So, by HL Postulate, $\Delta NQO\cong \Delta PQO$.

So, $\overline{NQ}\cong \overline{PQ}$.

Consider $\Delta NQM$ and $\Delta PQM$:

$\overline{NQ}\cong \overline{PQ}$

$\angle NQM\cong \angle PQM=90°$

$\overline{QM}\cong \overline{QM}$

So, by SAS Postulate, $\Delta NQM\cong \Delta PQM$.

So, $\overline{NM}\cong \overline{MP}$

Step 3.  State the conclusion.

Therefore, $\overline{NM}\cong \overline{MP}$ (proved).