Triangle Exterior Angle Theorem

Last updated on 26 October 2025

Table of contents

What is Exterior Angle Theorem?
Proof of the Exterior Angle Theorem
Solved Examples on Triangle Exterior Angle Theorem
FAQs on Triangle Exterior Angle Theorem

The Triangle Exterior Angle Theorem is a fundamental principle in geometry that relates to the relationship between the exterior angle of a triangle and the remote interior angles. This theorem is crucial in understanding the properties of triangles, particularly in relation to the exterior and interior angles.

What is Exterior Angle Theorem?

The Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote(opposite) interior angles.
In other words, if we extend one side of the triangle, creating an exterior angle at one of the vertices, the measure of this exterior angle is equal to the sum of the measures of the two interior angles that are not adjacent to it.

Proof of the Exterior Angle Theorem

Given: In △ABC in the figure above,

$∠ BAC$ , $∠ ABC$ and $∠ ACB$ are interior angles.
$∠ BAC = a$ , $∠ ABC = b$ , $∠ ACB = c$

Construction:

Extended side BC to point D to form an exterior angle $∠ ACD$ .
Consider $∠ ACD = x$

To Prove: We have to prove that the sum of the angles $∠ BAC$ and $∠ ABC$ is equal to $∠ ACD$ , i.e. $∠ ACD = ∠ BAC + ∠ ABC$

Proof: The sum of the angles that are formed on a straight line at the same point is always 180°.
Since, angles $∠ ACB$ and $∠ ACD$ are formed on the straight line segment BD at the same point C, the sum of these two angles is 180°. $\begin{array}{r} ∴ & ∠ ACB + ∠ ACD & = & 180 ° \\ (1) & ∴ & c + x & = & 180 ° \end{array}$
Also, Triangle's Angle Sum theorem states that the sum of all three interior angles of any triangle is always equal to 180°. $\begin{array}{r} ∴ & ∠ BAC + ∠ ABC + ∠ ACB & = & 180 ° \\ (2) & ∴ & a + b + c & = & 180 ° \end{array}$
Right hand sides of the equations $(1)$ and $(2)$ are equal.
Hence their left hand sides are also equal.
Therefore from equations $(1)$ and $(2)$ , $\begin{array}{lrcl} ∴ & c + x & = & a + b + c \\ ∴ & x & = & a + b + c - c \\ ∴ & x & = & a + b \\ ∴ & ∠ ACD & = & ∠ BAC + ∠ ABC \end{array}$
Hence proved the triangle exterior angle theorem.

Solved Examples on Triangle Exterior Angle Theorem

Example 1: Find the value of

x

in the triangle shown below.

Given: In the figure above,

Side QR is extended to point S.
Angles $∠ PQR$ , $∠ QPR$ and $∠ PRQ$ are interior angles of △PQR, and $∠ PRS$ is an exterior angle.
$∠ PRS = 135 °$ , $∠ PQR = 75 °$ and $∠ QPR = x$

Solution: By Exterior Angle Theorem, the measure of an exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles. $\begin{array}{lrcl} ∴ & ∠ QPR + ∠ PQR & = & ∠ PRS \\ ∴ & x + 75 ° & = & 135 ° \\ ∴ & x & = & 135 ° - 75 ° \\ ∴ & x & = & 60 ° \end{array}$ Hence, the value of $x$ is $60 °$ .

Example 2: In △ABC, an exterior angle at vertex A is 110°. The two opposite interior angles are in the ratio 2:3. Find all angles of the triangle.

Given:

Exterior angle = 110°
ratio of opposite interior angles = 2:3

Solution: Let the two opposite interior angles be $2 x$ and $3 x$ .
By the Triangle Exterior Angle Theorem, $\begin{array}{rcl} 2 x + 3 x & = & 110 \\ 5 x & = & 110 \\ x & = & \frac{110}{5} \\ x & = & 22 \end{array}$ So, the two opposite interior angles are: $2 x = 2 \times 22 = 44 °$ $3 x = 3 \times 22 = 66 °$ Now, the interior angle adjacent to the exterior angle is: $180 - 110 = 70 °$ Angles of the triangle: $44 °$ , $66 °$ , $70 °$

FAQs on Triangle Exterior Angle Theorem

What is the Triangle Exterior Angle Theorem?
The Triangle Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent (remote) interior angles.
How is the Triangle Exterior Angle Theorem mathematically expressed?
If △ABC has an exterior angle $∠ ACD$ formed by extending side BC, then:
$∠ ACD = ∠ CAB + ∠ ABC$
This shows that the exterior angle is equal to the sum of the two opposite interior angles.
Why is the Triangle Exterior Angle Theorem important?
It provides a way to relate different angles in a triangle, which can be useful in solving for missing angles and proving other geometric properties.
Does the Exterior Angle Theorem only apply to triangles?
Yes, the Exterior Angle Theorem specifically applies to triangles. However, understanding it can help in studying other polygons and their angle relationships.
Can I use the Triangle Exterior Angle Theorem with right triangles?
Yes, the theorem is applicable to all types of triangles, including right triangles, acute triangles and obtuse triangles.
Can the Triangle Exterior Angle Theorem help in real-life applications?
Yes, this theorem can be applied in architecture, engineering, and various fields that involve geometric calculations to determine angles and properties of structures.

Triangle Exterior Angle Theorem

What is Exterior Angle Theorem?

Proof of the Exterior Angle Theorem

Solved Examples on Triangle Exterior Angle Theorem

FAQs on Triangle Exterior Angle Theorem

What is the Triangle Exterior Angle Theorem?

How is the Triangle Exterior Angle Theorem mathematically expressed?

Why is the Triangle Exterior Angle Theorem important?

Does the Exterior Angle Theorem only apply to triangles?

Can I use the Triangle Exterior Angle Theorem with right triangles?

Can the Triangle Exterior Angle Theorem help in real-life applications?

Triangle Basic Proportionality Theorem

Triangle Inequality Theorem