![Two parallels and a transverse]()
Let MO and PR be two parallel lines and NQ is a transverse, that, on the way from point N to point Q, intersects, first, line MO at point L and then line PR at point K. Points P and O lie on one side of transverse PQ, while points M and R are on the other side.
Two one-sided angles (on one side of transverse NQ), of which one is interior or inner (between the parallels) and another exterior (outside of the area between the parallel lines), are called corresponding.
So, angles ∠MLN and ∠RKL are corresponding, so are some other pairs, like ∠NLO and ∠LKP etc. as shown on the picture by the same color.
Take into account that sum of measures of supplemental angles ∠MLN and ∠MLK is 180^o.
That is, ∠MLN + ∠MLK = 180^o or
∠MLN = 180^o - ∠MLK
Now, if the corresponding angles ∠MLN and ∠RKL are not equal in measure, then sum of inner angles /_MLK and /_RKL would be either less or greater than 180^o since /_MLN != /_RKL implies that ∠RKL != 180^o - ∠MLK and, therefore, ∠RKL + ∠MLK != 180^o.
There are two cases, both mean that a sum of some pair of inner angles measure less than 180^o:
∠RKL + ∠MLK < 180^o or
∠RKL + ∠MLK > 180^o, in which case, obviously, the sum of the other pair of inner angles' measures is less than 180^o:
∠PKL + ∠OLK < 180^o.
Let's recall the Fifth Postulate of Euclid.
If two lines on a plane intersect a third line in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.
Therefore, since ∠RKL + ∠MLK != 180^o, lines MO and RP are NOT parallel to each other, that contradicts the premise of the theorem.
