Triangle Angle Bisector Theorem

Last updated on 05 April 2025

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}

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.

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.

Triangle Angle Bisector Theorem

What is Angle Bisector Theorem?

Proof of the Triangle Angle Bisector Theorem

45-45-90 Triangle / Isosceles Right Triangle

Circle