Triangle Angle Bisector Theorem

Last updated on 01 November 2025

Table of contents

What is Angle Bisector Theorem?
Proof of the Triangle Angle Bisector Theorem
FAQs on Triangle Angle Bisector Theorem
Summary

The Angle Bisector Theorem is a fundamental concept in geometry that relates to the division of angles and sides within a triangle. This theorem provides a crucial understanding of the relationships between the lengths of the sides that form around the bisector of an angle within a triangle. It serves as a significant tool for exploring the properties of triangles and is essential for various geometric and trigonometric applications. Understanding the Angle Bisector Theorem is key to comprehending the intricacies of triangles and their constituent parts.

What is Angle Bisector Theorem?

The Angle Bisector Theorem states that in any given triangle, the angle bisector of a triangle divides the opposite side into two segments that are proportional to the lengths of the other two sides of the triangle.

Observe the above figure. In △PQR in the figure above, ray QM is the angle bisector of

∠ PQR

. Point M lies on side PR. The bisector QM divides the side PR in two segments, PM and MR. Therefore by Angle Bisector Theorem,

\frac{PQ}{QR} = \frac{PM}{MR} ⟹ \frac{a}{b} = \frac{x}{y}

Proof of the Triangle Angle Bisector Theorem

Given:

In $Δ LMN$ , Ray $LR$ is the bisector of $∠ MLN$ which intersects side $MN$ at point $R$ .
$LM = a$ , $LN = b$ , $MR = x$ , $RN = y$

To Prove: $\frac{a}{b} = \frac{x}{y}$

Construction:

Drawn ray $NS$ parallel to bisector $LR$ .
Extended the side $ML$ to intersect the ray $NS$ at point $S$ .

Proof: Since, ray $LR$ is the bisector of $∠ MLN$ , it divides the angle in two angles of equal measures. $\begin{matrix} (1) & ∴ ∠ MLR = ∠ RLN \end{matrix}$ Because $LR ∥ NS$ and $LN$ is the transversal, alternate angles are equal. $\begin{matrix} (2) & ∴ ∠ RLN = ∠ LNS \end{matrix}$ Also, ray $MS$ is the transversal to parallel lines $LR$ and $NS$ , hence corresponding angles are equal. $\begin{matrix} (3) & ∴ ∠ MLR = ∠ LSN \end{matrix}$ From $(1)$ , $(2)$ , and $(3)$ , we can say that, $∠ LNS = ∠ LSN$ It makes ΔLSN an isosceles triangle. Since sides opposite to equal angles are equal in an isosceles triangle, $\begin{matrix} (4) & ∴ LS = LN \end{matrix}$ Now, in ΔMSN, we have $LR ∥ NS$ .
By Thales' Theorem/Side Splitter Theorem, $\frac{ML}{LS} = \frac{MR}{RN}$ Put value of $LS$ from $(4)$ in above equation. $\begin{array}{lrcl} ∴ & \frac{ML}{LN} & = & \frac{MR}{RN} \\ ∴ & \frac{a}{b} & = & \frac{x}{y} \end{array}$ Hence proved the Triangle Angle Bisector Theorem.

FAQs on Triangle Angle Bisector Theorem

How is the Angle Bisector Theorem mathematically expressed?
If △ABC has a bisector of angle A that intersects side BC at point D, then:
$\frac{AB}{AC} = \frac{BD}{DC}$
This implies that the lengths of the segments on side BC are proportional to the lengths of sides AB and AC.
Can the Angle Bisector Theorem be applied to non-triangular shapes?
No, the Angle Bisector Theorem specifically applies to triangles. It does not extend to other polygon shapes.
How can the Angle Bisector Theorem be used in geometric proofs?
It can be used to demonstrate relationships in triangle properties, find missing side lengths, and establish ratios that lead to further conclusions about the triangle's properties.
Does the Angle Bisector Theorem apply to all types of triangles?
Yes, the Angle Bisector Theorem applies to all types of triangles, including scalene, isosceles, and equilateral triangles.
Is it possible to construct an angle bisector without measuring angles?
Yes, an angle bisector can be constructed using a compass and straightedge by creating arcs that intersect both rays of the angle, helping to find the bisector point.
What if I only know the lengths of the sides? Can I still use the Angle Bisector Theorem?
Yes, if you know the lengths of the sides, you can use the Angle Bisector Theorem to find the lengths of segments created on the opposite side by the angle bisector.
Are there any limitations to the Angle Bisector Theorem?
The theorem assumes that the triangle's sides are not degenerate (i.e., they do not collapse into a straight line). Further, the intersection of the bisector with the opposite side must be within the segment formed by that side.

Summary

Understanding the Angle Bisector Theorem is essential for gaining insights into the fundamental properties of triangles and angles within geometric shapes. Its applications extend beyond basic geometric concepts, making it a valuable tool in various fields of mathematics and sciences. The theorem's utility lies in its ability to provide precise relationships between angles and sides, facilitating accurate calculations and constructions in various mathematical contexts. Understanding this theorem is crucial for anyone delving into the realms of geometry and trigonometry, as it forms the basis for more advanced geometric and trigonometric concepts and applications.

Triangle Angle Bisector Theorem

What is Angle Bisector Theorem?

Proof of the Triangle Angle Bisector Theorem

FAQs on Triangle Angle Bisector Theorem

How is the Angle Bisector Theorem mathematically expressed?

Can the Angle Bisector Theorem be applied to non-triangular shapes?

How can the Angle Bisector Theorem be used in geometric proofs?

Does the Angle Bisector Theorem apply to all types of triangles?

Is it possible to construct an angle bisector without measuring angles?

What if I only know the lengths of the sides? Can I still use the Angle Bisector Theorem?

Are there any limitations to the Angle Bisector Theorem?

45-45-90 Triangle / Isosceles Right Triangle

Triangle Angle Sum Theorem