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In Geometry , two Lines or Planes (or a line and a plane), are considered perpendicular (or '''orthogonal''') to each other if they form Congruent Adjacent Angles . The term may be used as a Noun or Adjective . Thus, referring to Figure 1, the line AB is the perpendicular to CD through the point B. If a line is perpendicular to another as in Figure 1, all of the angles created by their intersection are called '' Right Angles '' (right angles measure ½ π Radian s, or 90 ° ). Conversely, any lines that meet to form right angles are perpendicular. The line AB does not have to end at B to be considered perpendicular. In a co-ordinate plane, perpendicular lines have opposite reciprocal slopes. Horizontal and vertical lines have zero and positive/negative infinity. NUMERICAL CRITERIA In terms of slopes In a Cartesian Coordinate System , two straight lines and may be described by equations. : : as long as neither is vertical. Then and are the Slopes of the two lines. The lines and are perpendicular if and only if the product of their slopes is -1, or if . The perpendiculars to vertical lines are always horizontal lines, and the perpendiculars to horizontal lines are always vertical lines. All horizontal lines are perpendicular to all vertical lines; that is, for any horizontal line and horizontal line , where and are constants, . CONSTRUCTION OF THE PERPENDICULAR To construct the perpendicular to the line AB through the point P using Compass And Straightedge , proceed as follows (see Figure 2).
To prove that the PQ is perpendicular to AB, use the SSS Congruence Theorem for triangles QPA' and QPB' to conclude that angles OPA' and OPB' are equal. Then use the SAS Congruence Theorem for triangles OPA' and OPB' to conclude that angles POA and POB are equal. IN RELATIONSHIP TO PARALLEL LINES c.]] As shown in Figure 3, if two lines (''a'' and ''b'') are both perpendicular to a third line (''c''), all of the angles formed on 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 Figure 3, 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:
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