Angle Bisector

Angle Bisector Definition

The definition of angle bisector says that an angle bisector is a line or ray that can divide an angle into two equal or congruent angles. For example, an angle of 70o when bisected by a line or right angle, the angle bisector is called a perpendicular bisector which leads to two 350 angles when bisected by a line or a ray. Angle bisector theorem has further established the derivation of this concept. The explanation of this theorem is discussed ahead. 

Exterior Angle Bisector Definition

The angle bisector defined above is the interior angle bisector. Likewise, there is an exterior angle bisector that is defined as a line or line segment that which divides into two congruent angles on the opposite side of the angle that is being bisected. In the case of a triangle, the bisector of the exterior angle divides or bisects the supplementary angle at a given vertex.

Angle Bisector Theorem

When the side lengths and angle bisector is known, angle bisector theorem can be applied then. According to the Angle Bisector Theorem, a triangle ABC, a line bisects the side BC at point D. When this line bisects the side BC, the ratio of BD to DC becomes equal to AB to AC. 


|AB| \ |AC| = |BD| / |DC|


Furthermore, when an external line AD bisects line BC, then the line AD is considered as the angle bisector of ∠A.


To conclude, the angle bisector theorem justifies that the angle bisector of the vertex angle will thereby bisect the opposite side of an isosceles triangle. 


Question


In the triangle ABC, AD bisects the side BC. The length of AB 6m, the length os AC is 12m and the length of DC is 8m. Find the length of the side BD?


Answer: 

As mentioned in the question above, this triangle is bisected by the line AD. Therefore, the formula of angle bisector theorem can be applied here.

|AB| \ |AC| = |BD| / |DC|


6/12 = BD/ 8

BD = 4m

Construction of an Angle Bisector 

The basic requirements for constructing an angle bisector are paper, pencil, eraser, ruler, and compass. With the help of these items, you can construct an angle bisector.


Follow the steps below when constructing an angle bisector.


Step 1:

The first step involves having an angle ∠ABC or ∠B that is given to you or drawn by you.

The image below represents the angle that will be bisected following the next steps.


Step 2:

After creating an angle, use a compass. The tip of the compass should be at point B and from there make an arc at each line. When you place the compass at point B, first make an arc at line AB and then another arc at line BC. Remember to do this without changing the position of the compass. 

The image below represents how these arcs will look like at each line


Step 3:

An important note here is not to change the radius of the compass. The radius that was maintained for marking the arc on line AB and BC, the same radius length will be used in this step as well. To make an intersecting arc, place the compass at the arc on line AB and draw one arc. Now place the compass at the arc on the line BC and make another arc that will cross the previous arc. 


This way you can create an intersecting arc which will lead to the last step for constructing an angle bisector. 


Step 4: 

The last step involved in the construction of an arc is a relatively simple one. For this step, take the help of a ruler. The intersecting arc when joined by a line to point B, will create a bisecting line that will further act as the bisector of the ∠ABC or ∠B.


The diagram below shows how the point of intersection between the arcs is connected with point B by a straight line using a ruler. 


In summary, follow these four steps for constructing an angle bisector. The first step is to create arcs on line AB and BC. This is followed by another step which involves intersecting arc. After you have formed the intersecting arc, you can then simply draw a straight line using a ruler to construct an angle bisector. 

FAQ (Frequently Asked Questions)

1.Is it possible to construct an angle bisector?

Yes. An angle bisector can be constructed with the help of a compass and ruler. Following the steps of forming arc on line segments, from which you can further construct an intersecting arc. This center point of this intersecting arc is then connected with the angle that gets bisected into two equal or congruent angles. It is also important to know the different kinds of angles to discover the angle bisector. Furthermore, the construction of angle bisector develops from the angle bisector theorem according to which the angle is bisected into two congruent angles. This construction method also justifies the derivation of the theorem which is:


|AB| / |AC| = |BD| / |DC|


2. Is the angle bisector theorem plausible?

The angle bisector theorem states that an angle bisector is a line or ray that can divide an angle into two equal or congruent angles. 

The proof of angle bisector theorem has been further derived through the law of sines. According to which, in two triangles (formed by a line segment of an angle bisector) ABC and ACD respectively, the derivation states:


|BD| / |DC| = |AB| Sin∠DAB /|AC| Sin∠DAC


Since the sin value of the two angles, Sin∠DAB and Sin∠DAC, is equal, the above-mentioned equation gets further simplified to :


|AB| / |AC| = |BD| / |DC|