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**