How to Find Angle in a Triangle?

A triangle is a three-sided closed polygon formed by the intersection of three lines. It is one of the basic shapes of geometry. It has three sides, three angles, and three vertices. A Right Angled Triangle is one where one of the angles is always equal to 90°.

Given the length of at least two sides of a Right-Angled triangle, we can find the value of any angle of the right-angled triangle. We use various trigonometric functions to find the angles in a triangle.

In this article, we will learn about, Angles in a Triangle, How to Find Angles in a Triangle, and others in detail.

Angles in a Triangle

In a triangle, there are three angles and these angles add up to 180°, i.e. sum of all the angles of a triangle are supplementary. The three interior angles in a triangle are used to define the angles in a triangle.

How to Find the Angle of a Triangle

To find the sum of the angles in a triangle we use the angle sum property in a triangle. We know that the sum of all the angles of a triangle adds up to 180°. And if any two angles of the triangle are given, we can easily find the third angle by adding the two angles and then subtracting the sum from 180°.

For example, if the two angles of a trinagle are 30° and 80° degrees then find the third angle of the triangle.

Let the third angle of the triangle is x, then

x + 30° + 80° = 180°

x + 110° = 180°

x = 180° - 110° = 70°

Thus, the third angle is 70°

How to Find the Missing Angle of a Triangle

To find the unknown angle ina triangle if any two angle of the traingle are given forrlow the following steps.

Step 1: Note the two unknown angles.

Step 2: Fine the sum of the unknown angles.

Step 3: Subtract the sum obtained in step 2 from 180°.

Step 4: The result obtained in step 3 is the required angle of the triangle.

Triangle Angle Formula

In a triangle ABC with angles ∠a, ∠b, and ∠c and sides AB, BC, and CA, triangle formula says that, sum of all the angles of the triangle is equal to 180°, i.e.

∠a + ∠b + ∠c = 180,

This formula is also called the Triangle Angle Sum Theorem.

Finding an Angle in a Right Angled Triangle

In a Right-angled triangle the unknown angle can also be found using various trigonometric functions depending on the sides of the right-angled triangel given.

Trigonometric Functions

In a right-angled triangle PQR,

PQ is the altitude of triangle, QR is the base of triangle, and PR is the hypotenuse of the triangle.

• cos θ: This gives the ratio of the base by the hypotenuse of a right-angled triangle.

cos θ = base / hypotenuse

• sin θ: This gives the ratio of altitude by the hypotenuse of a right-angled triangle.

sin θ = altitude / hypotenuse

• tan θ: It is the ratio of altitude by the base of a right-angled triangle.

tan θ = altitude / base

• cot θ: It is the reciprocal of tan θ
• sec θ: It is the reciprocal of cos θ
• cosec θ: It is the reciprocal of sin θ

Read, More

• How to Find Angle in a Right Angled Triangle
• Formula of finding Angles
• How to find Third Side of a Triangle with Two given Sides?

Examples on Angle Formulas

Example 1: Given a right-angled triangle, with base equals 10cm and hypotenuse equals 20cm. Find the value of the base angle.

Solution:

Given,

• Base = 10 cm
• Hypotenuse = 20 cm

Let, the base angle be θ.

cos θ = base / hypotenuse = 10/20 = 1/2

θ = cos^-1(1/2) = 60^o

Thus, the value of base angle is 60^o.

Example 2: Find the value of angles of a right angles triangle, given that one of the acute angles is twice the other.

Solution:

Sum of all the three angles in a triangle is 180^o.

Since one of the angles is 90^o and one of the acute angles is twice the other, we can consider them as θ and 2θ.

90^o + θ + 2θ = 180^o

3θ = 180^o - 90^o

3θ = 90^o

θ = 90^o/3 = 30^o

2θ = 2 × 30^o = 60^o

So, Angles in Triangle are 30^o, 60^o, and 90^o.

Example 3: Find the value of the angle of elevation of a ladder of length 5m, given that base of the ladder is at a distance of 3 m from the wall.

Solution:

Since the ladder acts as a hypotenuse of a right angles triangle and base distance equals 3 m, we can write

• Hypotenuse = 5 m
• Base = 3 m

Let the angle of elevation be θ. So, we can write

cos θ = Base / Hypotenuse = 3/5

θ = cos^-1(3/5)

θ = 53°

Thus, value of the angle of elevation is 53°.

Example 4: Find the value of hypotenuse, given the length of the altitude is 8m and the base angle equals 30°.

Solution:

Given, base angle is equal to 30^o and altitude equals 8m, we can apply the sine function to find the length of the hypotenuse.

sin 30° = Altitude / Hypotenuse

Hypotenuse = Altitude / sin 30°

(sin 30° = 1/2)

Hypotenuse = Altitude / (1/2) = 2 × Altitude

Thus, Hypotenuse = 2 × 8 = 16 m

Thus, the length of the hypotenuse is equal to 16 m.