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The Parallel & Perpendicular Postulates

Instructor: Samantha Dixon
Postulates are used in geometry to help prove theorems. This lesson explains how the parallel and perpendicular postulates will help to better understand the definition of parallel and perpendicular lines and how to prove theorems involving them.

Parallel and Perpendicular Lines

Parallel lines are straight lines that never intersect (cross each other). They are equidistant (the same distance) from each other. Arrows on both sides of the lines show that they continue infinitely. When graphing, parallel lines will have the same slope.

The joke above is a helpful way to remember the definition of parallel lines
Parallel

Perpendicular lines are straight lines that intersect to form a right (90-degree) angle. When graphing, perpendicular lines have opposite reciprocal slopes.

The two lines shown in this image are perpendicular. Right angles are usually indicated by a small square as seen in the top left angle
Perpendicular

Parallel and perpendicular lines are found in the same plane (space in math). Many theorems have been discovered about parallel and perpendicular lines. They are commonly introduced when learning about transversals, linear equations, and systems of equations.

Parallel Postulate

The parallel postulate states if there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line.

This image shows the parallel postulate using a point and line
Parallel postulate

The parallel postulate can be used to prove if lines are parallel to one another or not. It is important to remember that only one parallel line can be formed through the given point. The corresponding angles converse allows for a line to be constructed through the given point to make parallel lines. Given a line and a point not on that line, a parallel line can be drawn using a compass and ruler.

Perpendicular Postulate

The perpendicular postulate states if there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line.

This image shows the perpendicular postulate using a point and line
Perpendicular postulate

Similar to the parallel postulate, the perpendicular postulate can be used to prove if lines are perpendicular or not. Only one perpendicular line can be drawn through the given point. Given a line and a point not on that line, a perpendicular line can be drawn using a compass and ruler.

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