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