Angle Sum Property of a Triangle Explained

In this particular article, the topic of the Angle Sum Triangle has been explained in great detail using simple language and examples by the subject experts at Vedantu. Students are advised to download the free PDF of the notes and can also watch the video lectures on the topic to get a complete understanding and excel in their performance in any examination. 

What is a Triangle? 

Properties of Triangle

Interior and Exterior Angle of a Triangle

Angle sum property 

Exterior Angle property

Solved examples

What is a Triangle?

In our daily life, we all see a lot of things that are of different shapes and structures. Things like Traffic signs, pyramids, truss bridges, sailboards, and roofing a house are a few things that look like a triangle. So what is a triangle? A triangle is a closed polygon that is formed by only three line segments. Triangles can be classified based on their sides and angles. Based on their sides, triangles can be equilateral, isosceles, and scalene. And based on their angle, a triangle can be an acute triangle, right triangle, an obtuse triangle. 

Properties of Triangles

The properties of a triangle are listed below

• The sum of all the internal angles of a triangle is always equal to  180 degrees.

• The sum of the length of any two sides of a triangle is always greater than the third side of the triangle.

• The difference between the length of any two sides of a triangle is always less than the length of the third side of the triangle.

• The side that is at the opposite side of the greater angle is always the longest side among all the three sides of a triangle.

• In a triangle, the exterior angle is always equal to the sum of the interior opposite angle. This property is known as exterior angle property. 

• Any two triangles will be similar if their corresponding angles tend to be congruent and the length of their sides will be proportional.  

• The area of a triangle is ½ x base x height

• The perimeter of the triangle is the sum of all its three sides. 

Interior and Exterior Angle of a Triangle

There are two important attributes of a triangle i.e., angle sum property and exterior angle property. We may have questions in our minds about what is an angle sum property of a triangle? And how can we prove angle sum property? So let's clear our heads with these questions. 

We know already that a triangle has an interior as well as an exterior angle. The Interior angle is an angle between the adjacent sides of a triangle and an exterior angle is an angle between the side of a triangle and an adjacent side extending outward.

Angle Sum Property

Theorem: Prove that the sum of all the three angles of a triangle is 180 degrees or 2 right angles. 

Proof:  ∠1 = ∠B and ∠3 = ∠C………….(i)

Alternative angle = PQ||BC

∠1 +∠2 +∠3 = 180

∠B +∠2 + ∠C = 180

∠B +∠CAB + ∠C = 180

= 2 right angles 

Proved 

Theorem 2: In a triangle, if one side is formed then the exterior angle formed will be equal to the sum of two interior opposite angles.

 ∠4 = ∠1 + ∠2

Proof: ∠3 = 180 - (∠1 +∠2)..............(i)

∠3 + ∠4 = 180

Or ∠3 = 180 - ∠4…………(ii)

By (i) and (ii)

180 - (∠1 + ∠2) = 180 - ∠4

∠1 + ∠2 = ∠4

Proved

Exterior Angle Property

Proof of exterior angle property

The exterior angle theorem asserts that if a triangle’s side gets extended, then the resultant exterior angle will be equal to the total of the two opposite interior angles of the triangle.

According to the Exterior Angle Theorem, the sum of measures of ∠ABC and ∠CAB will be equal to the exterior angle ∠ACD. The general proof of this theorem is explained below:

Proof:

Consider  ∆ABC as given below such that the side BC of ∆ABC is extended. A parallel line to the side AB is drawn.

Thus, from the above statements, we can see that the exterior angle ∠ACD of ∆ABC is equal to the sum of two opposite interior angles i.e. ∠CAB and ∠ABC of the ∆ABC.

Solved Examples

Example 1) In the following triangle, find the value of x.

Solution)

x + 24 + 32 = 180

x + 56 = 180

x = 180 - 56

x = 124

Example 2) In the triangle ABC given below, find the area of a triangle inscribed inside a square of 20 cm.

Solution) Area of a triangle 

= \[\frac{1}{2} \times base \times height\]

= \[\frac{1}{2} (20) (20) \]

= \[200 cm^{2}\]

Key Learnings from the Chapter - 

• Students should be able to identify and label the three interior angles of any triangle 

• Remember that some of the interior angles of any triangle are equal to 180 degrees 

• There are two methods with which students can calculate the interior of any triangle