Angle Bisector Theorem

What is the Angle Bisector Theorem?

Angle bisector theorem is one of the most important theorems in geometry. The basic knowledge of a few terminologies is required to understand the angle bisector theorem. An angle is a separation between any two line segments. Bisect in English means to divide into two equal halves. An angle bisector is a line that divides an angle into two congruent parts. If the angle of separation between the two lines is 900, then the two lines are said to be perpendicular to each other. The perpendicular bisector of a line is that line which is perpendicular to the given line and divides the line into two identical halves.

Angle Bisector Theorem Statement:

The angle bisector theorem establishes a relationship between the lengths of the 2 sides of a triangle and the line segments formed when the angle bisector of the angle opposite to the third side is extended to meet the third side in a triangle. 

Angle Bisector theorem statement is as follows. “ In any triangle, when the angle bisector of one angle of a triangle is extended to meet the side opposite to that angle, the angle bisector divides the opposite sides in such a way that they are proportional to the other two sides of the triangle”. In other words, the angle bisector theorem statement can be rephrased as ‘if a point is on the angle bisector of an angle in a triangle, then the point is equidistant from the sides of the angle’.

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Explanation of what is Angle Bisector Theorem Converse:

In a triangle ABC as shown in the figure above, AD is the angle bisector of the angle A touching the side BC opposite to angle A. Then, according to the angle bisector theorem, AD divides BC in such a way that the quotient of BD and DC is equal to the quotient of AB and AC. It is written mathematically as:

\[\frac{{BD}}{{DC}} = \frac{{AB}}{{AC}}when\angle CAD\]

Converse of Angle Bisector Theorem Statement:

The converse statement of the angle bisector theorem says that if in a triangle ABC, D is a point on BC such that the ratio of BD and CD and that of AB and AC are the same, then AD is the angle bisector of A.

In other words, if an interior point is equidistant from the two sides of a triangle, then the point lies on the angle bisector of the angle formed by these two lines in any triangle. 

Explanation:

If D is a point in the interior of the triangle ABC such that it is at the same distance from the sides AB and AC, then the point D lies on the angle bisector of the angle A.

How to State and Prove Angle Bisector Theorem:

Angle bisector theorem is obtained by the law of sine or Sine rule in trigonometry. 

Angle Bisector Theorem Statement:

The angle bisector of one angle of a triangle divides the side opposite to it at a particular point such that the ratio in which the side is divided is equal to the ratio of the other two sides of the triangle.

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Given Data to State and Prove Angle Bisector Theorem:

ABC is a triangle with sides AB, BC, and AC. The angle bisector of A is AD which divides BC into BD and CD.

To Prove:

\[\frac{BD}{DC}\] = \[\frac{AB}{AC}\]


Angle Bisector Theorem Proof:

Statement

Reason

\[\frac{{BD}}{{\sin BAD}} = \frac{{AB}}{{\sin ADB}}\]

Applying Sine rule to the sides BD and AB in triangle ADB

\[\frac{{BD}}{{AB}} = \frac{{\sin BAD}}{{\sin ADB}} \to (1)\]

Rearranging the equation

\[\frac{{DC}}{{\sin DAC}} = \frac{{AC}}{{\sin ADC}}\]

Applying Sine rule to the sides DC and AC in triangle ADC

\[\frac{{DC}}{{AC}} = \frac{{\sin DAC}}{{\sin ADC}}{\text{ }} \to \left( 2 \right)\]

Rearranging the equation

Sin DAC  = Sin BAD → (3)

∠DAC = ∠BAD as AD is the angle bisector of ∠A

Sin ADB = Sin ADC → (4)

∠ADB and ∠ADC are supplementary angles as they form a linear pair. Sine of the supplementary angles are equal. 

\[\frac{{\sin DAC}}{{\sin ADC}} = \frac{{\sin BAD}}{{\sin ADB}} \to \left( 5 \right)\]

Dividing equation (3) by (4)

\[\frac{{BD}}{{AB}} = \frac{{DC}}{{AC}}\]

Comparing equations (1) and (2) with (5), RHS are equal and hence LHS should also be equal.

\[\frac{{BD}}{{CD}} = \frac{{AB}}{{AC}}\]

Rearranging the equation to obtain the desired result


This is the easiest method to arrive at the Angle bisector theorem proof. However, there are a few other complex methods to understand how to state and prove angle bisector theorems.

Angle Bisector Theorem Examples:

  1. If in a triangle ABC, AD is the angular bisector of ∠A which touches the side BC at D. Find AB and AC such that BD = 2 cm, CD = 5 cm, and AB + AC = 10 cm.

Solution:

The above problem is one of the angle bisector theorem examples. Applying angle bisector theorem to triangle ABC, we get 

\[\frac{{BD}}{{CD}} = \frac{{AB}}{{AC}}\]

It is given that BD = 2 cm and CD = 5 cm. If AB + AC = 10, then AC = 10 - AB. Substituting the values of AC, BD and CD in the above equation, we get 

\[\frac{2}{3} = \frac{{AB}}{{10 - AB}}\]

2 (10 - AB) = 5 AB

20 - 2 AB = 5 AB

20 = 5 AB + 2 AB

7 AB = 20 

AB = 20/7 cm

AC = 10 - AB

AC = 10 - 20/7

AC = 50/7 cm

Fun Facts:

  • Problems involving angle bisector theorem examples demand the basic knowledge of ratio and proportions.

  • Apart from using the sine rule, there are various other approaches to arrive at the Angle bisector theorem proof. 

FAQ (Frequently Asked Questions)

1. What is the Angle Bisector Theorem? 

Angle bisector theorem is a theorem in geometry proposed by Euclid, the father of geometry. This theorem gives the relation between the sides of the triangle and its angle bisector. The theorem states that the angle bisector of any angle in a triangle divides the side opposite to it such that the ratio in which the opposite side is divided is equal to the ratio of the other two sides which form the angle. If in a triangle ABC, AD is the angle bisector of A, then the angle bisector theorem states that the ratio of BD and BC is the same as the ratio of AB and AC. This can be proved by using the law of sines. 

BD/CD = AB/AC


2. What is Sine Rule? How is it used in Arriving at Angle Bisector Theorem Proof?

Sine rule or law of sines is a law in trigonometry which establishes a relationship between the sides and angles of a triangle. The rule states that the ratio of sine of the angle to its opposite side is the same as all the three angles and sides. In other words, the ratio of a side and the sine of the opposite angle remains the same for all the sides and angles of a triangle. If ABC is a triangle with side ‘a’ opposite to angle A, side ‘b’ opposite to the angle B and side ‘c’ opposite the angle C, then sine rule states that

a/Sin A = b/Sin B = c/Sin C

To obtain the angle bisector theorem proof, this sine rule is used to obtain the ratios of sides and angles in the two triangles formed by the angular bisector in a triangle.