Benchmark 24I can use properties of perpendicular bisectors and angle bisectors to identify equal distances.

Perpendicular Bisector Theorem

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

equidistant

Given: n is the bisector of ABProve: For any pt. P on n, PA=PB

C BA

P

Statements Reasons

1.Line n is the bisector of AB

1. Given

2. AC BC, 2. Def. of bisector

n

3. <ACP and <BCP are rt. <‘s

4. <ACP <BCP

3. Def. of

4. All rt. <‘s are

Given: n is the bisector of ABProve: For any pt. P on n, PA=PB

C BA

P

Statements Reasons

5. PC PC 5. Reflexive

6. ΔACP ΔBCP 6. SAS Post.

7. PA PB 7. CPCTC

8. PA = PB 8. Def. of segments

n

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

Converse of the Perpendicular Bisector Theorem

endpoints

Angle Bisector Theorem

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

equidistant

Converse Angle Bisector Theorem

If a point is on the ______ of an angle and is _________ from the sides of the angle, then it lies on the angle _______.

equidistantinterior

bisector