5-2 Perpendicular and Angle Bisectors

Objective

To use properties of perpendicular bisectors and angle bisectors

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In the Solve It, you thought about the relationships that must exist in order for a bulletin board to hang straight. You will explore these relationships in this lesson.

Essential Understanding There is a special relationship between the points on the perpendicular bisector of a segment and the endpoints of the segment.

In the diagram below on the left, C D ↔ modified c d with left right arrow above  is the perpendicular bisector of A B ¯ . C D ↔ eh b bar , . . modified c d with left right arrow above  is perpendicular to A B ¯ eh b bar  at its midpoint. In the diagram on the right, C A ¯ c eh bar  and C B ¯ c b bar  are drawn to complete Δ C A D cap delta c eh d  and Δ C B D . cap delta c b d .

You should recognize from your work in Chapter 4 that Δ C A D ≅ Δ C B D . cap delta c eh d approximately equal to cap delta c b d .  So you can conclude that C A ¯ ≅ C B ¯ , c eh bar , approximately equal to . c b bar , comma  or that C A = C B . c eh equals c b .  A point is equidistant from two objects if it is the same distance from the objects. So point C is equidistant from points A and B.

This suggests a proof of Theorem 5-2, the Perpendicular Bisector Theorem. Its converse is also true and is stated as Theorem 5-3.

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