This next set of proofs are all equivalent forms of 2.1 (Euclid's Fifth Axiom).

Theorem 2.1:Euclid’s Fifth Axiom:

That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

Theorem 2.2:Playfair’s Postulate:

For every line L and point P not on L there exists a unique line M that contains P and is parallel to L.

Theorem 2.3:Converse to AIA:

If L and M are parallel and N intersects both L and M then each pair of alternate interior angles are congruent.

Theorem 2.4

If a line intersects one of two parallel lines then it intersects the other.

Theorem 2.5:

If a line is perpendicular to one of two parallel lines then it is perpendicular to the other.

Theorem 2.6:Wallis’s Postulate:

Given any triangle ΔPQR and any line segment AB there exists a triangle ΔABC having AB as one of its sides such that ΔABC is similar to ΔPQR.

Theorem 2.7:

The angle sum of every triangle is 180.

Theorem 2.8

If L is parallel to M and M is parallel to N then L is parallel to N.

Theorem 2.9

Clairaut’s Axiom: Rectangles exist.

That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

Theorem 2.2:Playfair’s Postulate:

For every line L and point P not on L there exists a unique line M that contains P and is parallel to L.

Theorem 2.3:Converse to AIA:

If L and M are parallel and N intersects both L and M then each pair of alternate interior angles are congruent.

Theorem 2.4

If a line intersects one of two parallel lines then it intersects the other.

Theorem 2.5:

If a line is perpendicular to one of two parallel lines then it is perpendicular to the other.

Theorem 2.6:Wallis’s Postulate:

Given any triangle ΔPQR and any line segment AB there exists a triangle ΔABC having AB as one of its sides such that ΔABC is similar to ΔPQR.

Theorem 2.7:

The angle sum of every triangle is 180.

Theorem 2.8

If L is parallel to M and M is parallel to N then L is parallel to N.

Theorem 2.9

Clairaut’s Axiom: Rectangles exist.