page_title: Perpendicular
section_title: In relationship to parallel lines
hierarchical_section_title: Perpendicular / In relationship to parallel lines
context_page_description: In elementary geometry, the property of being perpendicular is the relationship between two lines which meet at a right angle. The property extends to other related geometric objects.
  A line is said to be perpendicular to another line if the two lines intersect at a right angle. Explicitly, a first line is perpendicular to a second line if the two lines meet; and at the point of intersection the straight angle on one side of the first line is cut by the second line into two congruent angles. Perpendicularity can be shown to be symmetric, meaning if a first line is perpendicular to a second line, then the second line is also perpendicular to the first. For this reason, we may speak of two lines as being perpendicular without specifying an order.
  Perpendicularity easily extends to segments and rays. For example, a line segment is perpendicular to a line segment if, when each is extended in both directions to form an infinite line, these two resulting lines are perpendicular in the sense above. In symbols, means line segment AB is perpendicular to line segment CD.
  A line is said to be perpendicular to a plane if it is perpendicular to every line in the plane that it intersects.
context_section_description: 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 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 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.
caption: The arrowhead marks indicate that the lines a and b, cut by the transversal line c, are parallel.