Editing Perpendicular (section)

== In relationship to parallel lines ==
[[File:perpendicular transversal v3.svg|class=skin-invert-image|thumb|236px<!--(as above)-->|The arrowhead marks indicate that the lines ''a'' and ''b'', cut by the [[transversal line]] ''c'', are parallel.]]

If two lines (''a'' and ''b'') are both perpendicular to a third line (''c''), all of the angles formed along the third line are right angles.  Therefore, in [[Euclidean geometry]], any two lines that are both perpendicular to a third line are [[parallel (geometry)|parallel]] to each other, because of the [[parallel postulate]].  Conversely, if one line is perpendicular to a second line, it is also perpendicular to any line parallel to that second line.

In the figure at the right, all of the orange-shaded angles are congruent to each other and all of the green-shaded angles are congruent to each other, because [[Vertical (angles)|vertical angles]] are congruent and alternate interior angles formed by a transversal cutting parallel lines are congruent.  Therefore, if lines ''a'' and ''b'' are parallel, any of the following conclusions leads to all of the others:

* One of the angles in the diagram is a right angle.
* One of the orange-shaded angles is congruent to one of the green-shaded angles.
* Line ''c'' is perpendicular to line ''a''.
* Line ''c'' is perpendicular to line ''b''.
* All four angles are equal.