Parallel Postulate (also known as Postulate 11 and the consequential Theorems 13 through 18 tell you that if two lines are parallel, then certain other statements are also true.

It is often useful to show that two lines are in fact parallel. For this purpose, you need theorems in the following form:

☛☛ If (certain statements are true) then (two lines are parallel).

It is important to realize that the converse of a theorem (the statement obtained by switching the if and then parts) is NOT always true. In this case, however, the converse of postulate 11 turns out to be true.

We state the converse of Postulate 11 as Postulate 12 and use it to prove that the converses of Theorems 13 through 18 are also theorems.

▶︎ POSTULATE 12:

If two lines and a transversal form equal corresponding angles, then the lines are parallel.

This postulate allows you to prove that all the converses of the previous theorems are also true.

▶︎ THEOREM 19:

If two lines and a transversal form equal alternate interior angles, then the lines are parallel.

▶︎ THEOREM 20:

If two lines and a transversal form equal alternate exterior angles, then the lines are parallel.

▶︎THEOREM 21:

If two lines and a transversal form consecutive interior angles that are supplementary, then the lines are parallel.

▶︎ THEOREM 22:

If two lines and a transversal form consecutive exterior angles that are supplementary, then the lines are parallel.

▶︎ THEOREM 23:

In a plane, if two lines are parallel to a third line, the two lines are parallel to each other.

▶︎ THEOREM 24:

In a plane, if two lines are perpendicular to the same line, then the two lines are parallel.

Copy and paste the following link into your browser to learn more about testing for parallel lines:

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