Triangle is the smallest polygon which has three sides and three interior angles. In this article, we are going to learn the interior angle sum property and exterior angle property of a triangle.

Interior Angle Sum Property of triangle

Theorem: The sum of interior angles of a triangle is 180° or two right angles (2x 90° )

Given: Consider a triangle ABC.

To Prove: ∠A + ∠B + ∠C = 180°

Construction: Draw a line PQ parallel to side BC of the given triangle and passing through point A.

Proof: Since PQ is a straight line, From linear pair it can be concluded that:

∠1 + ∠2+ ∠3 = 180° ………(1)

Since, PQ || BC and AB, AC are transversals

Therefore, ∠3 = ∠ACB (a pair of alternate angles)

Also, ∠1 = ∠ABC (a pair of alternate angles)

Substituting the value of ∠3 and ∠1 in equation (1),

     ∠ABC + ∠BAC + ∠ACB = 180°

⇒ ∠A + ∠B + ∠C = 180° = 2 x 90° = 2 right angles

Thus, the sum of the interior angles of a triangle is 180°.

Exterior Angle Property of Triangle

Theorem: If any one side of a triangle is produced then the exterior angle so formed is equal to the sum of two interior opposite angles.

Given: Consider a triangle ABC whose side BC is extended D, to form exterior angle ∠ACD.

To Prove: ∠ACD = ∠BAC + ∠ABC or, ∠4 = ∠1 + ∠2

Proof: ∠3 and ∠4 form a linear pair because they represent the adjacent angles on a straight line.

Thus, ∠3 + ∠4 = 180° ……….(2)

Also, from the interior angle sum property of triangle, it follows from the above triangle that:

∠1 + ∠2 + ∠3 = 180° ……….(3)

From equation (2) and (3) it follows that:

  ∠4 = ∠1 + ∠2

⇒ ∠ACD = ∠BAC + ∠ABC

Thus, the exterior angle of a triangle is equal to the sum of its opposite interior angles.

Note:

Following are some important points related to angles of a triangle:

1. Each angle of an equilateral triangle is 60°. 
2. The angles opposite to equal sides of an isosceles triangle are equal.
3. A triangle can not have more than one right angle or more than one obtuse angle.
4. In the right-angled triangle, the sum of two acute angles is 90°.
5. The angle opposite to the longer side is larger and vice-versa.

Solved Examples:

Q.1. Two angles of a triangle are of measure 600 and 450. Find the measure of the third angle.

Solution: Let the third angle be ∠A and the ∠B = 600 and ∠C = 450. Then, 

By interior angle sum property of triangles,
    ∠A + ∠B + ∠C = 1800
⇒ ∠A + 600 + 450 = 1800
⇒ ∠A + 1050 = 1800
⇒ ∠A = 180 -1050
⇒ ∠A = 750

So, the measure of the third angle of the given triangle is 750.

Q.2. If the angles of a triangle are in the ratio 2:3:4, determine the three angles.

Solution: Let the ratio be x.

So, the angles are 2x, 3x and 4x.
By interior angle sum property of triangle,
⇒ 2x + 3x + 4x =1800
⇒ 9x = 1800

⇒ x = 1800/ 9
⇒ x = 200
The three angles are:

2x = 2(200) = 400
3x = 3(200) = 600

4x = 4(200) = 800
So, the three angles of the triangle are 400, 600 and 800 respectively.

Q.3. Find the values of x and y in the following triangle.

Solution: Using exterior angle property of triangle,

     x + 50° = 92° (sum of opposite interior angles = exterior angle)
⇒ x = 92° – 50°

⇒ x = 42°

And, 

     y + 92° = 180° (interior angle + adjacent exterior angle = 180°.)
⇒ y = 180° – 92° 

⇒ y = 88°

So, the required values of x and y are 42° and 88° respectively