Perpendicular and Angle Bisectors

G.CO.9 Prove theorems about lines and angles... points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.

Goals for this section:

Students should be able to understand the perpendicular bisector theorem and the angle bisector theorem and how it could be applied to real-life situations.

Perpendicular Bisector Theorem

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

Let's break it down:

Notice the perpendicular bisector forms a right angle (perpendicular) with segment AB.
Notice the perpendicular bisector splits segment AB into two equal halves (bisects).
If you pick any point on the perpendicular bisector, A and B will be the same distance away from P (equidistant).

Extra Property!

The angles formed by MAP and MBP are congruent!
The triangle formed by the dotted lines (APB) is called an isosceles triangle.

Click here for examples on the Perpendicular Theorem

Angle Bisector Theorem

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

Let's break it down:

An angle bisector divides an angle in two equal halves.
If you pick any point P on the angle bisector, it will be the same distance away from the sides of the angle (equidistant) when you draw a perpendicular line to the sides.

Click here for an example using the Angle Bisector Example

External Resources

Perpendicular Bisector Theorem

Angle Bisector Theorem

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Keywords: perpendicular bisector theorem, angle bisector theorem, bisect, perpendicular

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