For contradiction, assume lines m and n are not parallel,  Then
they will intersect like this at point P, creating angle 9




Angle 1 is congruent to angle 8  (given)

Angle 8 is congruent to angle 5  (vertical angles) 

Angle 1 is congruent to angle 5  (angles congruent to the same angle are congruent)

Angles 3 and 1 are supplementary  (they form a straight angle)

Angles 3 and 5 are supplementary  (a supplement of a given angle is 
supplementary to an angle congruent to the given angle.)

Therefore measure of Angle 3 + measure of angle 5 = 180°

measure of Angle 9 + measure of Angle 3 + measure of Angle 5 = 180°
(the sum of the measures of the interior angles of a triangle is 180°)

Subtract equals from equals:
 
measure of Angle 9 + measure of Angle 3 + measure of Angle 5 = 180°  
                     measure of Angle 3 + measure of angle 5 = 180° 
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measure of Angle 9                                           = 0°

A triangle cannot have a 0° angle.

So the assumpion that limes m and n intersect is false.

Therefore the lines are parallel.

Edwin