Proofs of corresponding, alternate and cointerior angles

In plane geometry, proofs of properties to do with parallel lines (or their absence) rely ultimately on Euclid's Fifth Postulate. It asserts that: If a straight line crossing two straight lines [in a plane] makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which are the angles less than the two right angles.

Conversely, if the two lines do not meet, then the interior angles on the same side of a transversal must make two right angles. That is, they are supplementary. These angles are called co-interior angles.

In the diagram, we assume lines $BA$BA and $CD$CD extended indefinitely do not meet. Then, from the converse of the Fifth Postulate, we have 

$\angle ABC+\angle BCD=180^\circ$∠ABC+∠BCD=180°.

But, since $\angle EBC$∠EBC and $\angle ABC$∠ABC lie on the line $EBA$EBA, which makes two right-angles, we have

$\angle EBC+\angle ABC=180^\circ$∠EBC+∠ABC=180°. 

From this it is clear that $\angle EBC=\angle BCD$∠EBC=∠BCD  and we have proved the theorem that the interior alternate angles formed where a transversal crosses a pair of parallel lines are equal.

In a similar way, we can prove that the corresponding angles - angles in the same position relative to parallel lines crossed by another line - are equal. In this case, we use the fact that the two angles lying on the same side of the transversal where it crosses one of the parallel lines make two right-angles.

From the Fifth Postulate we have $\angle ABC+\angle BCD=180^\circ$∠ABC+∠BCD=180°. As before, $\angle ABC+\angle FBA=180^\circ$∠ABC+∠FBA=180°. Therefore, $\angle BCD=\angle FBA$∠BCD=∠FBA.

 

Examples

Example 1

If two lines in the plane meet and they are crossed by a third line, the three lines form a triangle and it follows from the Fifth Postulate that any two of the angles in the triangle must sum to less than two right angles.

In the diagram, lines $AB$AB and $AC$AC are not parallel. Therefore, according to Euclid's parallel postulate, $\angle ABC+\angle ACB<180^\circ$∠ABC+∠ACB<180°. The same argument can be applied for each pair of non-parallel lines.

It is interesting to note that while this is true in plane geometry, it is not necessarily true for triangles drawn on curved surfaces.

Example 2

Example 3