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.