**Theorem 6.2**: If a
line intersects two other
lines then the following
conditions are equivalent.

- a) The alternate interior angles are the same size
- b) The corresponding angles are the same size
- c) The opposite interior angles are supplementary.

**Proof**: If the alternate
interior angles are the same
size, then since one of
the corresponding angles is a
vertical angle to one of
the alternate interior angles,
and since by Theorem 5.11,
vertical angle are the
same size, the
corresponding angles are the
same size.

If the corresponding angles are the same size, then since one of the opposite interior angles is supplementary to one of the corresponding angles which is the same size as the other of the two opposite interior angles, the two opposite interior angles are supplementary.

If the opposite interior angles are supplementary, then since one of the alternate interior angles is supplementary to one of the opposite interior angles which is supplementary to the other of the two alternate interior angles, the alternate interior angles are the same size.