# 3.1: Equivalent Parallel Postulates

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Each of the following is an equivalent Euclidean postulate.

Equivalent Euclidean Postulates:

• (Playfair) Given a line and a point not on that line, there exists exactly one line through that point parallel to the given line.
• (Equidistance) Lines that are parallel are everywhere equidistant.
• (Euclid) Given two lines and a transversal of those lines, if the sum of the angles on one side of the transversal is less than two right angles then the lines meet on that side.

## 3.1.1Preparation

These theorems do not require a parallel postulate.

##### Theorem: Alternate Interior Angles

If the alternate interior angles formed by a transversal of two lines are equal, then the lines are parallel.

##### Theorem: Point to line distance

The (shortest) distance between a point and a line is from the point to the foot of the perpendicular.

##### Theorem: Convenient Euclid Parallel Axiom

Given two lines and a transversal of those lines, if the sum of the angles on one side of the transversal is equal to two right angles then the lines are parallel.

## 3.1.2Equivalency

The following theorem produces an easier to use version of Euclid's postulate.

The alternate interior angle converse theorem states Given parallel lines and a transversal of those lines, the alternate interior angles formed by the transversal are congruent.''

This page titled 3.1: Equivalent Parallel Postulates is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Mark A. Fitch via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.