# Exterior Angle Theorem of Triangle

According to the theorem, the exterior angle of a triangle is equal to the sum of two opposite interior angles.

Let us understand the theorem with the help of example.

In the below figure ABC is a triangle in which side BC is extended outside the triangle. Here ∠ABM is an external angle produced by external line BM and triangle side AB.

According to the external angle theorem;
∠ ABM = ∠A + ∠C

∠ ABM = External angle of triangle ABC
∠ A & ∠C = Opposite internal angles of ∠ ABM

I hope you understood the above concept. Now let us understand how to identify the exterior angle of any triangle.

### What is exterior angle of triangle ?

It is the angle formed by triangle’s side and its extension of adjacent side

The above image shows exterior angle ACD = 134 degree

Note the angle is formed by triangle side AC and extension of adjacent side BC

Here the external angle theorem work as:
∠ACD = ∠A + ∠B

### Proof of exterior angle theorem

Given below is triangle ABC in which side BC is extended towards left side to form exterior angle ∠ABM.

To Prove
∠x = ∠A + ∠C

Solution
ABC is a triangle
According to internal angle property
∠A + ∠B + ∠C = 180 degree —-eq(1)

Observe that MBC is a straight line
We known that in a straight line, the angles add up to make 180 degrees
∠x + ∠B = 180 degree — eq(2)

Comparing eq(1) and eq(2), we get:
∠A + ∠B + ∠C = ∠x + ∠B

Removing the common element;
∠A + ∠C = ∠x

Hence, we proved the exterior angle theorem of triangle.

### Prove that sum of all exterior angle of triangle add up to 360 degree

Solution
A figure of triangle is given above where:
a, b & c are the interior angle and d, e & f are the exterior angles

To prove
∠d + ∠e + ∠f = 360

Proof
Using exterior angle theorem, we know that:
∠d = ∠a + ∠b
∠e = ∠c + ∠b
∠f = ∠a + ∠c

Adding all the three equations, we get;
∠d + ∠e + ∠f = ∠a + ∠b + ∠c + ∠b + ∠a + ∠c
∠d + ∠e + ∠f = 2 (∠a + ∠b + ∠c) – – – -eq(1)

We know that sum of interior angle of triangle is 180 degree
∠a + ∠b + ∠c = 180

Putting the values in eq(1), we get;
∠d + ∠e + ∠f = 2 x 180
∠d + ∠e + ∠f = 360 degree

Hence Proved

### Problems on exterior angle theorem of triangle

Example 01
Find the value of exterior angle x

Solution
Using exterior angle theorem, we can write:
∠x = ∠A + ∠B
∠x = 50 + 80
∠x = 130 degree

Example 02
Given below is triangle ABC
BAM is an exterior angle formed by triangle side BA and extending alternate side CA

Solution
Using External Angle Theorem
125 = 78 + x
x = 125 – 78
x = 47 degree

Example 03
Write the exterior angle theorem of below triangle

Here ∠CAM = 115 is the exterior angle formed by side AC and extension of adjacent side BA

Applying external angle theorem, we get:
∠CAM = ∠1 + ∠2

Example 04
Write the exterior angle theorem of below triangle

BAM is another exterior angle formed by triangle side AB & extension of adjacent side CA.

According to the Exterior Angle Theorem
∠BAM = ∠1 + ∠2