**Theorem 6.4**: If two
lines are crossed by a third,
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.
- d) The two lines are parallel.

**Proof**: Theorem 6.2 says that a), b),
and c) are equivalent. Theorem 6.1 states that
d) implies a). So all that is needed is to show that c) implies d).

Suppose the two lines are not parallel. If they are not parallel, then they meet at a single point by Theorem 1.6. That point will be on one side of the transverse line or the other.

If the two lines cross,
then you will get a triangle, and the sum of the angles in the triangle will add up to 180^{o} by Theorem 6.3
so the two opposite interior
angles on the
side where the
two lines meet will have to
add up to less than 180^{o} by the number of degrees in the third angle.