Chapter 5Section 5.2

Perpendiculars and Bisectors

VocabularyPerpendicular Bisector: A segment, ray, line, or plane that is perpendicular to a segment at its midpoint is called a perpendicular bisector

Equidistant: Being the same distance away from two or more objects

A point can be equidistant from two other points

A point can be equidistant from two lines

Distance from a point to a line: Defined to be the length of a segment through the point perpendicular to the line

Perpendicular Bisector Theorem

Theorem

Theorem 5.2 Perpendicular Bisector Theorem

If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

CA = CB

is the perpendicular bisector of ABCD

Converse of the Perpendicular Bisector Theorem

Theorem

Theorem 5.3 Converse Perpendicular Bisector Theorem

If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of a segment.

CT = DT

T is on perpendicular bisector of CD

No, would need to know that there is a right angle

Yes, since CA = CB CBCA

Thus C is on the perpendicular bisector

Yes, it is possible to show that CA = CB

Angle Bisector Theorem

Theorem

Theorem 5.3 Angle Bisector Theorem

If a point is on the angle bisector of an angle, then it is equidistant from the two sides of the angle.

QR = SR

R is on the angle bisector of QPS

Converse of the Angle Bisector Theorem

Theorem

Theorem 5.4 Converse Angle Bisector Theorem

If a point is equidistant from the two sides of the angle, then it is on the angle bisector of an angle.

QR = SR

R is on angle bisector of QPS

No, need to know that P is equidistant to the rays (sides of the angle)

No, distance is measured perpendicularly

No, distance is measured perpendicularly

1. C is on the Bisector of AB 1. Given

DBAD.2 2. Definition Bisector

CBCA.3 3. Bisector Theorem

CDCD.4 4. Reflexive

5. ADC BDC 5. S.S.S.

1. WOZ WOY 1. Given

WOYWOZ;OYZO.2 2. Def. ’sXOZWOY;XOYWOZ.3 3. Vertical Angle Thm

XOZXOY.4 4. Transitive5. Reflexive XOXO.5

6. XOZ XOY 6. S.A.S.

XZXY.7 7. Def. ’s