Angle Bisector

An angle is formed by two different rays with the same initial point. The angle bisector is another ray sharing the same initial point.

Angle Bisector Definition


An angle bisector is a ray that divides an angles into two congruent angles.

Angle Bisector
 In the adjoining diagram ray OC divides $\angle$AOB into two congruent angles, $\angle$AOC and $\angle$BOC.

$\angle$AOC $\cong$ $\angle$BOC.

Now we can move on to angle Bisector theorem which states an important property of the angle bisector and the proof of the theorem.

Angle Bisector Theorem

Any point on the angle bisector is equidistant from the two arms of the angle.

This means, perpendiculars drawn to the arms from an arbitrary point on the angle bisector are equal in length

In the diagram given below perpendiculars DP and DQ are drawn from some point D on the bisector of the angle AOB to the arms OA and OB.

Angle Bisector Theorem

The angle bisector theorem states that DP $\cong$ DQ.

Angle Bisector Theorem Proof

Let D be a point on the angle bisector OC of triangle AOB.  Draw the perpendiculars DP and DQ to OA and OB.

Consider triangles ODP and ODQ

$\angle$POD $\cong$ $\angle$QOD - Given OD is the bisector of angle O.

$\angle$DPO $\cong$ $\angle$DQO - Right angles by construction

OD $\cong$ OD - Reflexive Property

$\triangle OPD \cong \triangle OQD$ - By AAS criteria for triangle congruence

DP $\cong$ DQ  CPCTC.

Hence it is proved D is equidistant from OA and OB

                                                                       Converse of angle bisector theorem
 If an interior point of an angle is equidistant from the two arms of the angle, then the point lies on the bisector of the angle.

Angle Bisector as Locus

In the following diagram, the BO is extended to form the angle AOC supplementary to the give angle AOB. OD is the bisector angle AOB and OE the bisector of angle AOC. It can be proved using the definitions of supplementary angles and angle bisectors, that two angle bisectors are perpendicular to each other.

Construct Angle Bisector

This property of the two angle bisectors of intersecting lines leads to the locus definition as follows:

The locus of a point equidistant from two intersecting lines is the pair of perpendicular lines that bisect the angles formed at the intersection of the lines.

Angle Bisector Construction


Construction Steps for Angle Bisector
 1. Equal arcs OC and OD are cut off OA and OB.

Angle Bisector Construction
 2. With C and D as centers equal arcs with equal radii are drawn. 
     Let the two arcs intersect at E.

 3. Join OE.

OE is the angle bisector of the given angle AOB.


Angle Bisector of a Triangle

Properties of angle bisectors of a triangle

The points of concurrency like circumcenter, orthocenter, incenter and centroid are important aspects of triangles. The angle bisectors of the triangle form one of these points.

The three angle bisectors of a triangle are concurrent. The point of concurrency is known as the incenter of the triangle.
 In the adjoining diagram the three angle bisectors of a triangle ABC, AI, BI and CI meet at I.  Triangle Angle Bisector
Since I is a point on the bisector of the angle A, I is equidistant from the sides AB and AC
of the triangle, and ID = IF.  By similar arguments, I is equidistant from AB and BC and ID = IE.

Hence I is equidistant from AB, BC and CD and ID = IE  = IF
The circle drawn with I as center and ID as radius will touch all the three sides of the triangle.
The circle is called the incircle of the triangle and I the point of concurrency of the angle bisectors
is known as the in center of the triangle.

Triangle Angle Bisector Theorem

An angle bisector in a triangle divides the opposite side in the same ratio as the sides containing the angle.

Triangle Angle Bisector Theorem
In the above diagram AD is the bisector of the angle A of triangle ABC.  It cuts the opposite side at D. According to angle bisector theorem,

$\frac{BD}{CD}=\frac{AB}{AC}$

In the above equation the numerators contain segments with common vertex B and the denominators contain segment with the common vertex C.

Triangle Angle Bisector Theorem Proof

Angle bisector theorem is proved with a construction and using side splitter theorem.

Given: AD is the bisector of angle A
Triangle Angle Bisector Theorem Proof 
To prove: $\frac{length \overline{BD}}{length\overline{CD}}= \frac{length\overline{AB}}{length\overline{AC}}$
Construction: Through C draw a line parallel to DA cutting BA extended at E.
                    The constructions done are shown with dotted lines.
 $\angle$BAD $\cong$ $\angle$AEC                   Corresponding angles by construction
     1   $\cong$   4
 $\angle$DAC $\cong$ $\angle$AEC                   Alternate interior angles by construction
     2    $\cong$  3
Angle 1 $\cong$ Angle 2              Given AD is the bisector of angle A.

Hence angle 3 $\cong$ angle 4     By substitution
 AC $\cong$ AE                           By congruence of isosceles triangle theorem.               
 In triangle BEC, AD || CE   By construction
 $\frac{BD}{DC}=\frac{BA}{AE}$     By Side splitter theorem.
 $\frac{BD}{DC}=\frac{BA}{AC}$     Substitution AC = AE
 Hence the angle bisector divides the opposite side BC in the ratio of the sides containing the angle.

The converse of the angle bisector theorem for triangles is also true.

Angle Bisector Theorem Examples

Below you could see examples for angle bisector theorem

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