Theorem 6.4: If two lines are crossed by a third, then the following conditions are equivalent.

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 180o 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 180o by the number of degrees in the third angle.

next theorem (6.5)