How to Find Angle in a Triangle?
Last Updated :
14 Dec, 2023
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.
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
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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) = 60o
Thus, the value of base angle is 60o.
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 180o.
Since one of the angles is 90o and one of the acute angles is twice the other, we can consider them as θ and 2θ.
90o + θ + 2θ = 180o
3θ = 180o – 90o
3θ = 90o
θ = 90o/3 = 30o
2θ = 2 × 30o = 60o
So, Angles in Triangle are 30o, 60o, and 90o.
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 30o 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.
FAQs
1. What is the Formula to Find Angles in a Triangle?
The unknown angle in a traingle is found using the Angle Sum Formula of Traingle. This formula states that some of all the angles of the triangle is 180°.
2. How do you Find the Angle of a Triangle with 3 sides?
The angles in a triangle if all its three sides are given is found using the cosine rule.
3. How do you Find the Missing Angle of a Triangle?
The missing angle of triangle is found using the angle sum property of the traingle or the trigonometric rules.
4. What is the Sum of Three Angles in a Triangle?
The sum of three angles in a triangle is 180°
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