Theorems around Perpendicular Lines

Another important concept is perpendicular. By definition, two lines room perpendicular if they crossing at ideal angles. That is, two perpendicular lines form 4 right angles. Segments and rays can likewise be perpendicular. This way they intersect in at the very least one point, and also the 2 lines containing them space perpendicular.

You are watching: Two lines perpendicular to a third line are parallel.

We use perpendicular segment to measure the distance from a point to a line, a allude to a plane, or the distance between two parallel lines or planes. The ties the a railroad track space perpendicular to the rails and of the very same length. This common length is the distance in between the rails. (If parallel present exist, then railroad tracks in space can go on forever.)

There room three theorems around perpendicular lines that you have to know. We will certainly not attempt to prove them here, yet if friend think around them they should be fairly obvious:

Theorem 1: provided a line and also a suggest not on the line, there is one and also only one line with the given allude that is perpendicular to the given line.


We have the right to use this fact to specify the distance from a point to a line: That street is the length of a segment perpendicular to the line through the given suggest as one of its endpoints and also the other endpoint ~ above the line. In fact, a similar notion holds in 3 dimensions. If we have a aircraft and a allude not on that plane, climate there is just one line through the point perpendicular to the plane, and also the size of the segment established by that point and the intersection the the perpendicular line through the airplane is identified as the street from the allude to the plane.

Theorem 2: If 2 coplanar lines space perpendicular to a 3rd line, then they room parallel to each other:


This is a special situation of the an ext general an outcome that when two coplanar currently are cut by a transversal and corresponding angles room equal, climate the lines are parallel. In this case, the matching angles are ideal angles.

See more: Is 57 A Prime Or Composite Number, 57 (Number)

Theorem 3 (the Perpendicular Bisector Theorem):

Any point on the perpendicular bisector that a segment is equidistant from the endpoints of the segment, and also conversely:


The perpendicular bisector the a segment is a line that goes with the midpoint of the segment and is perpendicular to it. This organize tells united state we have the right to think that a perpendicular bisector as the set of all points that are equidistant native the endpoints that the segment.