Angle Bisector Theorem

Angle Bisector Theorem
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"The kite rises highest against the wind, not with it."

Isn't this an inspiring thought?

Yes, indeed!

Let's have a closer look at the kite. Are you able to see angles being formed at each of its corners?

The kite rises highest against the wind, not with it.

In the figure, we can observe that segment BD has divided \(\angle B\) and  \(\angle D\) into equal parts. 

That's called an angle bisector.

In this short lesson, we will be focusing more on the most important theorem on the angle bisector. To be more precise, we will be learning about the angle bisector theorem proof, angle bisector theorem examples, triangle angle bisector theorem, how to construct angle bisector, and other interesting properties and facts around angle bisectors.

Lesson Plan

What Is Angle Bisector Theorem?

Triangle angle bisector theorem states that "In a triangle, the angle bisector of any angle will divide the opposite side in the ratio of the sides containing the angle".

Consider the figure below.

Statement of Angle Bisector Theorem

Here, AD is the bisector of \(\angle A\).

According to the angle bisector theorem, \(\dfrac{BD}{DC}=\dfrac{AB}{AC}\).

Construction Of Angle Bisector

Use the simulation below to visualize the steps of construction for angle bisector.


Proof Of Angle Bisector Theorem

Angle Bisector Theorem Proof

To prove the angle bisector theorem, we will do construction.

Draw a ray CX parallel to AD, and extend BA to intersect this ray at E

Proof of Triangle angle bisector theorem

By the Basic Proportionality Theorem, we have that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.

In \(\Delta CBE\), DA is parallel to CE.

\(\dfrac{BD}{DC}=\dfrac{BA}{AE}\;\;\;\;\;\; \cdots (1)\)

Now, we are left with proving that AE = AC.

Let's mark the angles in the above figure.

Proof of Triangle angle bisector theorem

Since DA is parallel to CE, we have

  1. \(\angle 1=\angle 2\) (corresponding angles)
  2. \(\angle 3=\angle 4\) (alternate interior angles)

Since AD is the bisector of \(\angle BAC\), we have \(\angle 1=\angle 3\).

So, we can say that \(\angle 2=\angle 4\).

Since sides opposite to equal angles are equal, AC = AE.

Substitute AC for AE in Equation (1).

\(\dfrac{BD}{DC}=\dfrac{BA}{AC}\)

Hence proved.

 
important notes to remember
Important Notes
  1. An angle bisector is a line that divides the angle into two equal angles.

  2. The triangle angle bisector theorem is an important consequence of the Basic Proportionality Theorem.

  3. Any point on the bisector of an angle is equidistant from the sides of the angle.

Solved Examples

Example 1

 

 

Amy drew a triangle on the board.

She asks if \(\dfrac{AB}{AC}=\dfrac{BD}{DC}\).

Ms. Amy drew a triangle on the board and asks a question to students

Can you answer this?

Solution

Let's find the ratio \(\dfrac{AB}{AC}\).

\[\begin{align}\dfrac{AB}{AC}=\dfrac{4}{6}=\dfrac{2}{3}\end{align}\]

Let's find the ratio \(\dfrac{BD}{DC}\).

\[\begin{align}\dfrac{BD}{DC}=\dfrac{1.6}{2.4}=\dfrac{2}{3}\end{align}\]

\(\therefore\) Both the ratios are equal.
Example 2

 

 

In \(\Delta XYZ\), XE is the bisector of \(\angle X\).

 XE is the bisector of angle X

Let XY = 4 units, YE = 2 units, and EZ = 3 units.

Can you find the length of XZ?

Solution

Given that, XE is the bisector of \(\angle X\).

So, we can use the angle bisector theorem.

According to the question, 

\[\begin{align}\dfrac{YE}{EZ}&=\dfrac{XY}{XZ}\\\dfrac{2}{3}&=\dfrac{4}{XZ}\\XZ&=\dfrac{4}{2}\times 3\\XY&=6\end{align}\]

\(\therefore\) The length of XZ is 6 units.
Example 3

 

 

Look at \(\Delta ABC\) shown below.

AD is the bisector of angle A

If AD bisects \(\angle A\), can you find the value of \(x\)?

Solution

Given that, AD is the bisector of \(\angle A\).

So, we can use the angle bisector theorem.

According to the question, 

\[\begin{align}\dfrac{AB}{AC}&=\dfrac{BD}{DC}\\\dfrac{x}{x-2}&=\dfrac{x+2}{x+1}\\x(x-1)&=(x-2)(x+2)\\x^2-x&=x^2-4\\-x&=-4\\x&=4\end{align}\]

\(\therefore\) The value of \(x\) is 4.
Example 4

 

 

Jenny drew a right-angled triangle as shown.

Jenny asked Jack to determine the value of x

She asked Jack to determine the value of \(x\).

Can you help him?

Solution

In \(\Delta ABC\), \(\angle DBC=45^{\circ}\).

Since the triangle is right-angled at \(\angle B\), we can say that BD bisects the angle \(ABC\).

By triangle angle bisector theorem, \(\dfrac{AB}{BC}=\dfrac{AD}{DC}\)

Substitute \(AB=5\), \(BC=12\), \(AD=3.5\), and \(DC=x\).

\[\begin{align}\dfrac{AB}{BC}&=\dfrac{AD}{DC}\\\dfrac{5}{12}&=\dfrac{3.5}{x}\\5x&=42\\x&=8.4\end{align}\]

\(\therefore\) The length of \(x\) is 8.4 units.
 
Thinking out of the box
Think Tank
  1. Can you use properties of similarity of triangles to prove the triangle angle bisector theorem?
  2. Can you find the length of an angle bisector in a triangle? What is the minimum data required to find the length of an angle bisector?

Interactive Questions

Here are a few activities for you to practice.

Select/type your answer and click the "Check Answer" button to see the result.

 
 
 
 

Let's Summarize

We hope you enjoyed learning about the triangle angle bisector theorem with the examples and practice questions. Now, you will be able to easily solve problems on the triangle angle bisector theorem.

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FAQs on Triangle Angle Bisector Theorem

1. What is the formula for angle bisector?

Let AD be the bisector of \(\angle A\) in \(\Delta ABC\).

Proof of Triangle angle bisector theorem

According to the angle bisector theorem, \(\dfrac{BD}{DC}=\dfrac{AB}{AC}\).

2. How are the side-splitter theorem and the angle bisector theorem similar?

The only similarity between the side-splitter theorem and the angle bisector theorem is that both the theorems related the proportions of side lengths of the triangle.

3. What is the converse of the angle bisector theorem?

If AD is drawn in the \(\Delta ABC\) such that \(\dfrac{BD}{DC}=\dfrac{AB}{AC}\), then AD bisects the \(\angle A\).

Proof of Triangle angle bisector theorem

4. How do you find the angle bisector of a triangle?

Follow the steps mentioned below to bisect \(\angle PQR\).

  1. Let Q be the center and with any radius, draw an arc intersecting the ray \(\overrightarrow{QP}\) and \(\overrightarrow{QR}\), say at the points E and D respectively.
  2. Now, taking D and E as centers and the same radius, draw arcs intersecting each other say at F. <<DE is not a line. Pls check. Also, I guess the radius should be same as the step 1>>
  3. Draw the ray \(\overrightarrow{QF}\).

OF is the bisector of angle PQR

Here, \(\overrightarrow{QP}\) is the angle bisector of \(\angle PQR\).

5. How do you solve a triangle bisector?

We solve a triangle bisector theorem by using the basic proportionality theorem.

6. How do you solve an angle bisector problem?

We solve an angle bisector problem by using the triangle angle bisector theorem.

7. When can you use the angle bisector theorem?

The angle bisector theorem is used when we know the angle bisector and length of sides of the triangle.

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