Triangle Angle Bisector Theorem

Angle Bisectors & Perpendicular Bisectors

Angle Bisectors

Bisecting an angle means dividing it into two congruent angles.

Solution:

Each new angle measures 15°. Multiply by 2 to find the original, larger angle.

15° × 2 = 30°

So, the original angle measured 30°.

Multiple Choice

The figure shows a triangle with one of its angle bisectors. Find x if m∠2 = 4x + 5 and m∠1 = 5x − 2.

• x = 7

• x = 3

• x = -7

• x = 1/3

Angle Bisector Theorem

The Angle Bisector Theorem states that a point is on the bisector of an angle if and only if it is in the interior of the angle and is equidistant from the sides of the angle.

Z is on the angle bisector of ∠WXY if and only if it is in the interior of ∠WXY and WZ

≅YZ

.

Example:

Look at the Diagram. Since m∠GHJ=m∠IHJ=66°, 

HJ bisects ∠GHI.

Therefore, GJ≅IJ

 by the Angle Bisector Theorem.

So, I=G=49.

Perpendicular Bisector Theorem

The Perpendicular Bisector Theorem 

states that a point is on the perpendicular bisector of a segment if and only if it is equidistant from the endpoints of the segment.

B is on the perpendicular bisector of AC if and only if AB≅BC

.

Example

Since UV=VW=54, V is equidistant from the endpoints of UW. Therefore, V lies on the perpendicular bisector of UW by the Perpendicular Bisector Theorem.

Since V is on the perpendicular bisector of UW and ∠VXW is a right angle, VX is the perpendicular bisector of UW.

So, UX=WX=35 and UW=2(35)=70.

Solution: Label the diagram

Since ∠TSU is a right angle and S=S=65, SU is the perpendicular bisector of TV.

Therefore, TU≅UV

 by the Perpendicular Bisector Theorem.

So, U=T=98.

Multiple Choice

Can you use the perpendicular bisector theorem to solve for GH?

• Yes.

• No. We haven't been given enough information.