Let’s talk about the Angle Bisector Theorem, a cool concept in geometry that can really come in handy, especially on tests like the SAT, ACT, and AP exams. The Angle Bisector Theorem tells us that if you draw an angle bisector in a triangle (basically a line that cuts an angle into two equal halves), it divides the opposite side of the triangle in such a way that the two parts are proportional to the other two sides of the triangle. Sounds complicated? It’s actually pretty straightforward once you break it down!

To understand this, picture a triangle with two sides—let's call them "a" and "b." The angle bisector divides the opposite side into two parts that are proportional to "a" and "b." The idea is that the length of these segments (let's call them "x" and "y") follows the ratio of the two sides. You’ll see this come up in questions about triangles on exams like the SAT, ACT, and even the GRE, so knowing this theorem will definitely give you a boost!

Now, let’s break down some key terms to make sure everything’s clear. First, an angle is formed when two lines intersect at a specific point, and we represent angles with the symbol ∠. An angle bisector is a line that divides an angle into two equal smaller angles. So, when you extend a ray that bisects an angle, it’s dividing it into two parts with the same measure—pretty neat, right?

In this article, we’ll go over the Angle Bisector Theorem, how to prove it, its formula, types, and even its converse. Whether you're prepping for the SAT, AP, or even the MCAT, this concept is definitely something you’ll want to have in your toolkit!

Angle Bisector Theorem

Bisector means the elements that bisect a given shape or an object into two identical parts and an angle is developed when two straight lines or rays meet at a common endpoint.

The angle bisector theorem says that an angle bisector of a triangle divides/splits the opposite side of the triangle into two elements such that they are proportional to the other two sides of the triangle. Or you can understand this as, by the application of the angle bisector theorem, the opposite side of a triangle will be divided into two proportional parts to the triangle’s other two sides.

Here, for the triangle ΔABC, the line CD is the bisector for the angle C. Thus according to the theorem;

\(\frac{AD}{DB}=\frac{AC}{CB}\)

The theorem can be applied to all types of triangles, be it equilateral triangle, isosceles triangle, scalene triangle, right-angled triangles, etc.

Learn about the different types of Quadrilaterals here.

Types of Angle Bisector Theorem

As of now, we know what is angle bisector theorem through its definition. Let us understand the two different types of angle bisector theorems.

• Interior Angle Bisector Theorem
• Exterior Angle Bisector Theorem

Interior Angle Bisector Theorem

The internal angle bisector in the given triangle divides the opposite side internally in the ratio of the sides including the vertical angle.

Consider the below image, here for the triangle ABC, AD is the internal bisector that meets BC at D and internally bisects the ∠BAC. We will discuss the proof in the coming header.

Exterior Angle Bisector Theorem

The external angle bisector in the given triangle splits the opposite side externally in the ratio of the sides holding the angle.

Consider the below image, here for the triangle ABC, AD is the external bisector of the angle BAC which meets BC at point D. We will discuss the proof below.

Angle Bisector Theorem Formula

As per the statement, we can say that the internal or external bisector of an angle in a triangle divides or splits the opposite side internally or externally in the ratio of the corresponding angles and sides having the angle.

For the diagram below in the triangle ABC, AD is the bisector, then the angle bisector theorem formula is;\(\frac{BD}{DC}=\frac{AB}{AC}\)

Angle Bisector Theorem Proof

Now that we know the internal as well as the external angle bisector theorem with the formula, let us understand the proof for the same.

Internal Angle Bisector Theorem Proof

Theorem 1 : For the triangle ΔABC, we can say that AD is the internal bisector for the ∠BAC which intersects BC at point D.

According to the theorem we need to prove that;\(\frac{BD}{DC}=\frac{AB}{AC}\).

Step 1 : Mark a line CE that is parallel to AD mathematically saying CE ∥ DA. Now extend a line from E such that it meets point A as shown in the figure above.

Step 2 : For the diagram above we can say that CE is parallel to DA also AC is transversal. The alternate interior angles initiated by a transversal are identical.

• Thus ∠DAC=∠ACE (as these are alternate angles)…….. (equation-1)
• Also, ∠BAD=∠AEC (as there are corresponding angles)………. (equation-2)

Step 3 : We know that AD is the angle bisector of A,

• ∠BAD=∠DAC………(equation-3)

Step 4 : From equations 2, and 3 we can say that;

• ∠AEC=∠DAC………(equation-4)

Step 5 : Now comparing equation 4, and 1 we can say that;

• ∠ACE=∠AEC………(equation-5)

Step 6 : Thus if we consider the triangle ACE, where ∠ACE=∠AEC, implies that it is an isosceles triangle.

• Also, being an isosceles triangle the side AE is equal to the side AC.

Step 7 : Now applying the proportionality theorem to the triangle BCE:

Theorem 2 : When a line is marked parallel to one side of the provided triangle and intersects the other two sides at different points, then the other two sides are divided in the exact or same ratio.

Thus; for the triangle BCE, CE is parallel to DA.

Step 1 : By the proportionality theorem;

• \(\frac{BD}{DC}=\frac{AB}{AE}\)

Step 2 : As we know, AE=AC, then the ratio modified to;

• \(\frac{BD}{DC}=\frac{AB}{AC}\) Hence proved.

Proof of External Angle Bisector Theorem

Similar to the internal angle proof, let us try to proof the formula for the external angle as well. Here for the triangle ABC, AD act as the external bisector for the angle BAC that intersects BC at point D.

We need to prove: \(\frac{BD}{DC}=\frac{AB}{AC}\)

Step 1 : Here we have drawn CE parallel DA such that C meets E on the line AB.

• As CE is parallel to DA and AC is a transversal, we are having;
• ∠ECA = ∠CAD (these are alternate angles) …….. (equation-1)

Step 2 : Similarly, CE is parallel to DA and now we take BP as the transversal, we have;

• ∠CEA = ∠DAP (the angles are corresponding angles) ……..(equation-2)

Step 3 : Also, we have AD as the bisector of ∠CAP;

• ∠CAD = ∠DAP ……..(equation-3)

Step 4 : From equations 2 and 3 we have:

• ∠CEA=∠CAD ……..(equation-4)

Step 5 : Now compare equations 1 and 4.

• ∠CEA=∠ECA
• Hence the triangle EAC is an isosceles triangle. And in an isosceles triangle, two sides are equal. Here AE=AC.

Step 6 : Thus if we consider the triangle BDA, EC is parallel to AD. Applying the proportionality theorem to the triangle we get;

• \(\frac{BE}{EA}=\frac{BC}{CD}\)

Step 7 : Add 1 to both sides of the equation.

• \(\frac{BE}{EA}+1=\frac{BC}{CD}+1\)

Step 8 : Simplifying the expression we get;

• \(\frac{\left(BE+EA\right)}{EA}=\frac{\left(BC+CD\right)}{CD}\)

Step 9 : On further simplification we have;

• \(\frac{AB}{EA}=\frac{BD}{CD}\)

Step 10 : As we know, AE=AC, then the ratio modified to;

• \(\frac{BD}{DC}=\frac{AB}{AC}\) Hence proved.

Converse of Angle Bisector Theorem

The converse of the angle bisector theorem states that if a given triangle satisfies the below condition;

Condition: If a line sketched from one vertex of the triangle splits the opposing side into two portions such that they are symmetrical to the rest two sides of the triangle.

Conclusion: If the above statement is true then the point on the opposite side of the angle will lie on the angle bisector.

If \(\frac{BD}{DC}=\frac{AB}{AC}\), then D will lie on the angle bisector of angle A or AD is the angle bisector of angle A.

In other words in a triangle, when the interior point is at an equal distance from the two sides of a triangle then that point will lie on the angle bisector of the angle created by the two line segments.

If D is a point in the interior of angle BAC. Also, if the perpendicular bisector distances DC and DB are equal then, the line AD will act as the angle bisector of angle ∠BAC. Thus ∠BAD will be equal to ∠CAD.

Solved Examples on Angle Bisector Theorem

The angle bisector theorem enables us to determine the unknown lengths of sides of triangles as the angle bisector separates the side opposite to the angle into two segments such that they are proportional to the triangle’s other two sides. With all this information below are some practice questions, try to solve them on your own and check your answer.

Solved Example 1: For the given triangle ABC if AD is the angle bisector for angle A. Obtain the value of x?

Solution: Given, AD is the angle bisector for angle A. Using the angle bisector theorem formula.

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

AB=20, AC=16, BD=x, DC=4.

Substituting the values in the equation we get;

\(\frac{x}{4}=\frac{20}{16}\Rightarrow\ x=5\).

Solved Example 2: For the given triangle PQR, QS is the angle bisector. Answer the questions for the below statements.

Statement 1:SA is equal to____

Statement 2:∠BQS is equal to_____

Solution: From the diagram, we can state that;

As QS is the angle bisector, SA is equal to SB.

Also, ∠BQS is equal to ∠SQR.

Solved Example 3: Consider the ΔABC, If BD is the angle bisects for angle B, determine the value of x?

Solution: Given, that BD is the bisector for the angle B.

AB=4, BC=6, AD=x, DC=x+2.

Applying the angle bisector formula;

\(\frac{AB}{BC}=\frac{AD}{DC}\)

\(\frac{4}{6}=\frac{x}{x+2}\)

Solving the equation we get;

6x=4x+8

2x=8

x=4

Thus the value of x=4.

Conclusion

In conclusion, the Angle Bisector Theorem is a powerful tool in geometry, especially for exams like the SAT, ACT, and AP. It helps us understand how an angle bisector divides the opposite side of a triangle into two proportional parts. By mastering this theorem, including its formula, types, and proofs, you’ll be ready to tackle related questions in your exams with confidence. Keep practicing, and you’ll definitely ace those geometry problems!