Angle Bisector Theorem: Definition & Theorems of Triangle

Angle Bisector Theorem states that an angle bisector is the other side of the triangle so that the ratio of the two line segments is equal to the ratio of the other two sides. As a result, the lengths of the other two triangle sides are equal to the relative lengths of the opposite side (divided by the angle bisector). A line that divides an angle into two congruent halves is known as an angle bisector. The two lines are considered to be perpendicular to one another if the angle of separation between them is 90 degree. The line that is perpendicular to the provided line and splits it into two identical halves is the perpendicular bisector of the identical line. All kinds of triangles can be used to prove the angle bisector theorem.

Read more: Types of angles

KeyTerms: Angle bisector theorem, Converse bisector theorem, Interior angle, External angle, Perpendicular bisector theorem

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What is Angle Bisector Theorem?

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Angle bisectors are lines that divide an angle into two equal or congruent angles and are drawn from the vertex of a triangle to its opposite side.The two fundamental characteristics the points on an angle bisector are evenly spaced from each other and divide a triangle's opposing side proportionately to its adjacent sides, that is regarded as the triangle's angle bisector property.

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Angle Bisector Theorems of Triangle

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The internal and external angle bisector theorem- related statements are listed in the table below.

According to the angle bisector theorem, the opposite side of a triangle is divided into two portions by the angle bisector that are proportional to the other two sides.

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Interior Angle Bisector Theorem

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The angle bisector of the triangle ABC intersects side BC at point D. As mentioned in the picture below.

Interior Angle Bisector Theorem

According to angle bisector theorem, the ratio of the line segment BD to DC equals the ratio of the length of the side AB to AC

\(|\frac{BD}{DC}| = |\frac{AB}{AC}|\)

In contrast, when a point D on the side BC divides BC in a ratio similar to the sides AC and AB, then the angle bisector of ∠ A is AD. Hence, according to the theorem, if D lies on the side BC, then,

\(|\frac{BD}{DC}| = |\frac{AB}{AC}| \)\(\frac{sin \angle DAB}{sin \angle DAC}\)

If D is external to the side BC, directed angles and directed line segments are required to be applied in the calculation.

It is possible to use the angle bisector theorem when side lengths and angle bisectors are known.

Proof of Angle bisector theorem

Angle bisector theorem can be proved using trigonometry. In triangles ABD and ACD using the law of sines, can be written as;

\(\frac{AB}{BD} = \frac{sin \angle BDA}{sin \angle BAD}\)……..(1)

\(\frac{AC}{DC} = \frac{sin \angle ADC}{sin \angle DAC}\)……..(2)

The angles ∠ ADC and ∠ BDA make a linear pair and hence called supplementary angles. 

Since, the sine of supplementary angles are equal, therefore, 

Sin ∠ BDA = Sin ∠ ADC …………(3)

Also,

∠DAC = ∠BAD (AD is the angle bisector)

Thus, 

Sin ∠BDA = Sin ∠ADC ………(4)

Hence, from equations 3 and 4, the RHS of equations 1 and 2 are equal, therefore, LHS will also be equal.

\(\frac{|BD|}{|DC|} = \frac{|AB|}{|AC|}\)

Hence, the angle bisector theorem is proved. 

Condition: 

If the angles ∠DAC and ∠BAD are not equal, the equation 1 and equation 2 can be written as: 

\(\frac{|AB|}{|BD|}\)sin ∠ BAD = sin ∠ BDA

\(\frac{|AC|}{|DC|}\)sin ∠ DAC = sin ∠ ADC

Angles ∠ ADC and ∠ BAD are supplementary, hence the RHS of the equations are still equal. Hence, we get

\(\frac{|AB|}{|BD|}\) sin ∠ BAD = \(\frac{|AC|}{|DC|}\) sin ∠ DAC

This rearranges a generalized view of the theorem. 

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Converse of Angle Bisector Theorem

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The converse bisector theorem of an angle states that if a triangle's interior point is equally spaced from its two sides, that point will be on the angle's bisector, which is the angle produced by the two line segments.

Triangle Angle Bisector Theorem

Triangle Angle Bisector Theorem

Extend the side CA to meet BE to meet at point E, such that BE//AD.

Now we can write,

CD/DB = CA/AE (since AD//BE) —-(1)

∠4  =  ∠1 [corresponding angles]

∠1 = ∠2 [AD bisects angle CAB]

∠2 = ∠3 [Alternate interior angles]

∠3 = ∠4 [By transitive property]

ΔABE is an isosceles triangle with AE=AB 

Now if we replace AE by AB in equation 1, we get;

CD/DB = CA/AB

Hence proved.

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Perpendicular Bisector Theorem

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This theorem states that a point is on the perpendicular bisector of a line segment if it is equally spaced from the line segment's endpoints in a triangle.

Alternatively, it can be stated that the perpendicular bisector divides the provided line segment, to which it is perpendicular, into two equal portions. If a perpendicular bisector is drawn from a triangle's vertex to the other side, it separates the segment into two congruent segments.

Perpendicular Bisector Theorem

As represented in the above figure, the line segment SI is the perpendicular bisector of WM. 

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External Angle Bisector Theorem

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The opposite side of a triangle is divided externally by the external angle bisector in the ratio of the sides that contain the angle. Most frequently, non-equilateral triangles exhibit this property.

Proof:

Given: In ΔABC, The external bisector of ∠BAC is, which intersects BC created at D. 

To prove: BD/DC = AB/AC

Construction: Draw CE ∥ DA meeting AB at E

External Angle Bisector Theorem

Since, CE ∥ DA and AC is transversal, therefore, 

∠ECA = ∠CAD (alternate angles) ……(1)

Again, CE ∥ DA and BP is a transversal, therefore,

∠CEA = ∠DAP (corresponding angles) —–(2)

But AD is the bisector of ∠CAP, 

∠CAD = ∠DAP —–(3)

As we know, Sides opposite to equal angles are equal, therefore,

∠CEA = ∠ECA

In ΔBDA, EC ∥ AD. 

BD/DC = BA/AE [By Thales Theorem]

AE = AC, 

BD/DC = BA/AC

Hence, proved.

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Things to remember

• Angle Bisector theorem states that the angle bisector of a triangle bisects the opposite side in such a way that the ratio of the two lines segments is proportional to the ratio of the other two sides. 
• Angle bisector theorem is applicable to all types of triangles.
• The angle bisector theorem includes interior angles and exterior angles. 
• The angle bisector theorem formula is as follows: \(|\frac{BD}{DC}| = |\frac{AB}{AC}|\)

• According to the converse of the internal angle bisector theorem, the interior point of a triangle is on the angle bisector of the angle created by the two line segments if it is equally distant from the two sides of the triangle.
• The provided line segment is split in half by the perpendicular bisector into two equal parts.