# Properties of angle of triangle

In this chapter, we will discuss important properties of angle of triangle with examples.

These properties are important as they help to solve different geometry related problems.

## Important angle property of triangle

(a) The sum of all angles of triangle add to 180 degree.

Given above is the triangle ABC with interior angles \mathtt{\angle 1,\ \angle 2\ \&\ \angle 3} .

According to the property;

\mathtt{\angle 1\ +\angle 2\ +\ \angle 3\ =\ 180\ degree}

This property is also called angle sum property of triangle.

(b) The measure of exterior angle in triangle is equal to sum of opposite angles.

Given above is the triangle ABC with exterior angle ∠4.

According to exterior angle property of triangle;

\mathtt{\angle 4=\ \angle 1+\ \angle 2}

(c) A triangle can have only one 90 degree angle.

If one of the angle measure 90 degree then the other two angles will definitely be less than 90 degree.

Given above is the triangle with ∠B = 90 degree.

Since one of the angle measure 90 degree, the other two angles will be acute angles.

(d) In triangle with 90 degree angle, the sum of other two acute angle will also be 90 degree.

Taking the same example as above, the triangle given is ABC with ∠B = 90 degree then sum of other two angle will also be 90 degree.

\mathtt{\Longrightarrow \ \angle A+\angle C}\\\ \\ \mathtt{\Longrightarrow \ 60\ +\ 30}\\\ \\ \mathtt{\Longrightarrow \ 90\ degree}

(e) The sum of adjacent interior and exterior angle of triangle measure exactly 180 degree.

In the above triangle ABC, ∠ACB & ∠ACD are the adjacent interior and exterior angles. If you add both these angles, you will get 180 degree.

⟹ ∠ACB + ∠ACD

⟹ 65 + 115

⟹ 180 degree.

(f) Triangles with equal sides have equal angle opposite to them.

In the above triangle ABC; side AB = AC = 4 cm.

If sides are equal then the angle opposite to them are also equal.

Hence, ∠1 = ∠ 2 = 45 degree.

(f) Angles in scalene, isosceles and equilateral triangle

Scalene triangle
In scalene triangle, all sides are different so all angles are also different to each other.

Isosceles Triangle
In isosceles triangle, two sides are equal to each other so the opposite angles are also equal.

Hence, in isosceles triangle two angles are equal to each other.

Equilateral triangle
In equilateral triangle, all side lengths are equal so all angle are equal to each other.

I hope you understood the above properties, let us solve some problems related to the concept.

## Angles of Triangle – Solved examples

Example 01
Given below is the triangle ABC. Find the measure of ∠ACB.

Solution
The angle sum property says that sum of all interior angle of triangle measure 180 degree.

∠A + ∠B + ∠C = 180

70 + 45 + ∠C = 180

115 + ∠C = 180

∠C = 180 – 115

∠C = 65 degree

Hence, ∠ACB measures 65 degree.

Example 02
In the below triangle ABC, find the measure of angle x.

Solution
According to exterior angle property, the measure of exterior angle is equal to sum of opposite interior angle.

∠x = ∠ABC + ∠ACB

∠x = 55 + 40

∠ x = 95 degree

Hence, the angle x measures 95 degrees.

Example 03
Analyze the below figure, check if it is scalene, isosceles or equilateral triangle.

Solution
Note that ∠B = ∠C = 50 degree.

We know that sides opposite to equal angles are also equal.
So AB = AC.

Hence, the figure is isosceles triangle.