5.1.1 Perpendicular and Angle Bisector Theorems Key Objectives • Prove and apply theorems about perpendicular bisectors. • Prove and apply theorems about angle bisectors. Key Terms • When a point is the same distance from two or more objects, the point is said to be equidistant from the objects. • A locus is a set of points that satisfies a given condition. • An angle bisector is a ray that divides an angle into two congruent angles. • The perpendicular bisector of a segment is a line that passes through the midpoint of a segment that is perpendicular to the segment. Theorems, Postulates, Corollaries, and Properties • Perpendicular Bisector Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. • Converse of the Perpendicular Bisector Theorem If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. • Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. • Converse of the Angle Bisector Theorem If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle. Example 1 Using Perpendicular Bisector Theorems
By the Perpendicular Bisector Theorem, if a point is on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment.
By the Converse of the Perpendicular Bisector Theorem, if a point is equidistant from the endpoints of a segment, then the point is on the perpendicular bisector of the segment.
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