3.1.4 Properties of Perpendicular Lines Key Objectives • Prove and apply theorems about perpendicular lines. Key Terms • The perpendicular bisector of a segment is a line perpendicular to a segment at the segment’s midpoint. • The distance from a point to a line is the length of the perpendicular segment from the point to the line. Theorems, Postulates, Corollaries, and Properties • Linear Pair of Congruent Angles Postulate If two intersecting lines form a linear pair of congruent angles, then the lines are perpendicular. • Perpendicular Transversal Theorem In a plane, if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other line. • Lines Perpendicular to the Same Line Theorem If two coplanar lines are perpendicular to the same line, then the two lines are parallel to each other. Example 1 Distance from a Point to a Line The distance from a point to a line is used in this example to write and solve an inequality.
Notice that three line segments from the point Q to TR are shown in this figure, QT , QS , and QR . The line segment that is also perpendicular to TR , QS , is the distance from Q to TR . Therefore, QS must be the shortest distance from Q to TR . It follows that QS < QT . Substitute the given expression for QS , x + 3 and the given length for QT , 10, into the inequality, QS < QT , and solve for x .
Example 2 Proving Properties of Lines
By the Perpendicular Transversal Theorem, in a plane, if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other line.
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