Exterior Angle Property of a Triangle Theorem

Theorem 2: If any side of a triangle is extended, then the exterior angle so formed is the sum of the two opposite interior angles of the triangle.

Proof:

 

As we have proved the sum of all the interior angles of a triangle is 180° (∠ACB + ∠ABC + ∠BAC = 180°) and we can also see in figure, that ∠ACB + ∠ACD = 180° due to the straight line. By the above two equations, we can conclude that 

∠ACD = 180° – ∠ACB

⇒ ∠ACD = 180° – (180° – ∠ABC – ∠CAB)

⇒ ∠ACD = ∠ABC + ∠CAB

Hence proved that If any side of a triangle is extended, then the exterior angle so formed is the sum of the two opposite interior angles of the triangle.

Example: In the triangle ABC, ∠BAC = 60° and ∠ABC = 70° then find the measure of angle ∠ACB.

Solution:

The solution to this problem can be approached in two ways:

Method 1: By angle sum property of a triangle we know ∠ACB + ∠ABC + ∠BAC = 180°

So therefore ∠ACB = 180° – ∠ABC – ∠BAC

⇒ ∠ACB = 180° – 70° – 60°

⇒ ∠ACB = 50°

And ∠ACB and ∠ACD are linear pair of angles, 

⇒ ∠ACB + ∠ACD = 180° 

⇒ ∠ACD = 180° – ∠ACB  = 180° – 50° = 130°

Method 2: By exterior angle sum property of a triangle, we know that ∠ACD = ∠BAC + ∠ABC

∠ACD = 70° + 60°

⇒ ∠ACD = 130°

⇒ ∠ACB = 180° – ∠ACD

⇒ ∠ACB = 180° – 130°

⇒ ∠ACB = 50° 

Read More about Exterior Angle Theorem.

Angle Sum Property of a Triangle is the special property of a triangle that is used to find the value of an unknown angle in the triangle. It is the most widely used property of a triangle and according to this property, “Sum of All the Angles of a Triangle is equal to 180º.”

Angle Sum Property of a Triangle is applicable to any of the triangles whether it is a right, acute, obtuse angle triangle or any other type of triangle. So, let’s learn about this fundamental property of a triangle i.e., “Angle Sum Property “.