Given that;
[tex]\begin{gathered} \measuredangle MOP\text{ is a right angle.} \\ \measuredangle MOP=90^0 \end{gathered}[/tex]
And;
[tex]\vec{RP}\perp\vec{OP}[/tex]
Since line RP is perpendicular to line OP, Angle RPO must be a right angle.
[tex]\measuredangle RPO=90^0[/tex]
Recall that for two parallel lines intersected by a straight line, Same side interior angles are supplementary.
[tex]A+B=180^0[/tex]
So, for line MO to be parallel to line RP, the sum of angle MOP and angle RPO must be equal to 180 degree.
[tex]\measuredangle MOP+\measuredangle RPO=90+90=180^0[/tex]
Since the sum of angle MOP and angle RPO is equal to 180 degree, then line MO is parallel to line RP.
[tex]\begin{gathered} \text{ Since} \\ \measuredangle MOP+\measuredangle RPO=180^0 \\ \text{Then;} \\ MO\Vert RP \end{gathered}[/tex]
Proved
