Angle Sum Property of a Triangle

Triangle is the smallest polygon which has three sides and three interior angles, consisting of 3 edges and 3 vertices. A triangle with vertices A, B and C is denoted as ∆ABC. In a triangle, 3 sides and 3 angles are referred to as the elements of the triangle. Angle sum property and exterior angle property are the two important attributes of a triangle. 

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.

Angle Sum Property of A Triangle‌

A triangle is the smallest polygon. It has three interior angles on each of its vertices. Triangles are classified on the basis of

Interior angles as an acute-angled triangle, obtuse-angled triangle and right-angled triangle.

Length of sides as an equilateral triangle, isosceles triangle and scalene triangle.

A common property of all kinds of triangles is the angle sum property. The angle sum property of triangles is 180°. This means that the sum of all the interior angles of a triangle is equal to 180°. This property is useful in calculating the missing angle in a triangle or to verify whether the given shape is a triangle or not. It is also frequently used to calculate the exterior angles of a triangle when interior angles are given. For example,

In a given triangle ABC,

∠ABC + ∠ACB + ∠CAB = 180°

When two interior angles of a triangle are known, it is possible to determine the third angle using the Triangle Angle Sum Theorem. To find the third unknown angle of a triangle, subtract the sum of the two known angles from 180 degrees.

Let’s take a look at a few example problems:

Example 1

Triangle ABC is such that, ∠A = 38° and ∠B = 134°. Calculate ∠C.

Solution

By Triangle Angle Sum Theorem, we have;

∠A + ∠B + ∠C = 180°

⇒ 38° + 134° + ∠Z = 180°

⇒ 172° + ∠C = 180°

Subtract both sides by 172°

⇒ 172° – 172° + ∠C = 180° – 172°

Therefore, ∠C = 8°

Solved Examples:

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.

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.

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