Question 424126
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{{{drawing(1000,200,-20,20,-8,8,
line(-19,602/163,4,602/163), line(-19,-602/159,4,-602/159),line(2,7,3,-7),
locate(1.7,5.1,1), locate(1.8,3.7,3), locate(4,4,m),locate(4,-3.8,n),
locate(2.3,5,2), locate(2.4,3.7,4),
locate(3,-3.8,8),locate(1.7+.6,5.1-7.5,5), locate(1.8+.6,3.7-7.5,7),
locate(2.3+.6,5-7.5,6)
 )}}}


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


{{{drawing(1000,200,-20,20,-8,8,locate(1.7,5.1,1), 
locate(1.7,5.1,1), locate(1.8,3.7,3),locate(4,4,m),locate(4,-4,n),
locate(-19,1,P), locate(-15,.5,9),
locate(2.3,5,2), locate(2.4,3.7,4),locate(3,-3.8,8),locate(1.7+.6,5.1-7.5,5), locate(1.8+.6,3.7-7.5,7),
locate(2.3+.6,5-7.5,6),


line(-19,0,4,4), line(-19,0,4,-4),line(2,7,3,-7) )}}}

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° 
-------------------------------------------------------------------
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</pre>

Angle 3 + Angle 5 = 180° (they are supplementary.


Angles 5 and 7 are supplementary  (they form a straight angle)

 
Angle 1 is congruent to angle 5