Last Updated : 19 Jul, 2024

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Angle Sum Property of a triangle is a fundamental idea in geometry, which states that the sum of the internal angles of a triangle is always 180 degrees.

In this article, we have covered Angle Sum Property of a Triangle, its Practice Worksheet and others in detail.

## What is Angle Sum Property of a Triangle?

Angle Sum Property of a Triangle** , **states that the sum of all internal angles of a triangle is equal to 180

^{o}. Whether a triangle is scalene, isosceles, or equilateral, this feature applies to all of them. Knowing this feature is essential for resolving a wide range of geometric issues and establishing increasingly intricate theorems.

The sum of all internal angles of a triangle is equal to 180^{o}.

Angle Sum Property of a Triangle is used to:

- Calculate the unknown angles, if we know the other angles in a triangle.
- Confirm that a shape is indeed a triangle.

## Important Formulas / Concepts

The exterior angle theorem states that the measurements of a triangle’s two non-adjacent inner angles add up to the outer angle. Mathematically it can be represented as:

**ExteriorAngle = Opposite Angle1 + Opposite Angle2****Sum of All Angles of Triangles = 180°**

## Practice Worksheet Angle Sum Property of a Triangle

**Q1. In triangle DEF, if ∠D = 40° and ∠E = 100°, find ∠F. **

**Q2. In triangle GHI, if ∠G = 35° and ∠H = 65°, find ∠I.**

**Q3. In triangle JKL, if ∠J = 55° and ∠K = 85°, find ∠L.**

**Q4. In triangle MNO, if ∠M = 30° and ∠N = 100°, find ∠O.**

**Q5. In triangle ABC, if ∠A = 50° and ∠B = 60°, find ∠C.**

**Q6. In triangle XYZ, if ∠X = 80° and ∠Y = 70°, find ∠Z.**

**Q7. In triangle PQR, if ∠P = 90° and ∠Q = 45°, find ∠R.**

**Q8. Find the measure of all angles in an equilateral triangle.**

**Q9. In a triangle, A = 45**^{o}** and B = 75**^{o}**. Find angle C.**

**Q10. A triangle has angles A and B such that A=3B. If angle C=30**^{o}**, find angles A and B.**

**Q11. In an isosceles triangle, the two equal angles are 50**^{o}**. Find the third angle.**

**Q12. Find the measure of each angle in an equilateral triangle.**

## Angle Sum Property of a Triangle Examples with Solution

**Example 1: In triangle XYZ , if ∠X = 50**^{o}** , ∠Y = 60**^{o}** , find ∠Z. **

**Solution:**

Using the angle sum property,

∠X + ∠Y + ∠Z = 180

^{o}∠Z = 180

^{o}– (∠X + ∠Y)∠Z = 180

^{o}– ( 50^{o}+ 60^{o})∠Z = 180

^{o}– (110^{o})∠Z = 70

^{o}

**Example 2: In triangle ABC , if ∠A = 100**^{o}** , ∠B = 30**^{o}** , find ∠C. **

**Solution:**

Using the angle sum property,

∠A + ∠B + ∠C = 180

^{o}∠C = 180

^{o}– (∠A + ∠B)∠C = 180

^{o}– ( 100^{o}+ 30^{o })∠C = 180

^{o}– (130^{o})∠C = 50

^{o}

**Example 3: In triangle PQR , if ∠P = 35**^{o}** , ∠Q = 65**^{o}** , find ∠R. **

**Solution:**

Using the angle sum property,

∠P + ∠Q + ∠R = 180

^{o}∠R = 180

^{o}– (∠P + ∠Q)∠R = 180

^{o}– (35^{o}+ 65^{o})∠R = 180

^{o}– (100)∠R = 80

^{o}

**Example 4: In a given triangle ABC, if all angles are equal. Find measure of each angle. **

**Solution:**

Using the angle sum property,

∠A + ∠B + ∠C = 180

^{o }(

)Given all angles are equalLet’s suppose ∠A = ∠B = ∠C = x

^{o }Now ,

x + x + x = 180

^{o}3x = 180

^{o}x = 60

^{o}Measure of each angle=60

^{o}.All angles are equal to 60

^{o}. This is an equilateral triangle.

**Example 5: In an isosceles triangle STU, angles S and T are equal, and angle U = 40 degrees. Find angles S and T.**

**Solution:**

Using the angle sum property,

Let’s suppose ∠S = ∠T = x

^{o}∠S + ∠T + ∠U = 180

x + x + 40 = 180 (Given, ∠U=40)

2x + 40 = 180

2x = 140

x = 70

So, angles ∠S and ∠T are 70 degrees each

## Frequently Asked Questions

### Why is Angle Sum Property Significant?

The angle sum property is commonly used in many geometric proofs and problem solving situations, and it is essential for comprehending geometry’s fundamental ideas.

### Is it Possible to Apply Angle Sum Property to Every Triangle?

,it is possible to apply the angle sum property to every triangle, whether a triangle is acute, obtuse, or right-angled, the angle sum characteristic still holds true.Yes

### Is it Possible to Determine Unknown Angles in a Triangle using Angle Sum Property?

Yes, when the other angles in the triangle are known, the angle sum feature is frequently employed to find unknown angles.

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