In this article, we are going to learn about the simplest form of a polygon, * a triangle*. All polygons can be divided into triangles, or in other words, they are formed by combining two or more triangles. Thus, understanding the basic properties of a triangle and their types is essential.

Here is an outline of the topics we will cover in this article:

- Definition of a Triangle
- Triangles: Classification by Type
- Special Cases of Right-Angled Triangles
- Triangle Formula: Area
- Properties of Triangle: Summary and Key Takeaways
- Triangle Properties Application Quiz

You can also view this video on the properties of triangle:

## What is a triangle?

As the name suggests, the *triangle* is a polygon that has three angles. So, when does a closed figure has three angles?

When it has three line segments joined end to end.

Thus, we can say that a triangle is a polygon, which has three sides, three angles, three vertices and the sum of all three angles of any triangle equals 180°.

## Properties of a triangle

These are the properties of a triangle:

- A triangle has three sides, three angles, and three vertices.
- The sum of all internal angles of a triangle is always equal to 180
^{°. }This is called the angle sum property of a triangle. - The sum of the length of any two sides of a triangle is greater than the length of the third side.
- The side opposite to the largest angle of a triangle is the largest side.
- Any exterior angle of the triangle is equal to the sum of its interior opposite angles. This is called the exterior angle property of a triangle.

## Types of triangles

Triangles can be classified in 2 major ways:

- Classification according to internal angles
- Classification according to the length of its sides

### Classification of a triangle by internal angles

Based on the angle measurement, there are three types of triangles:

- Acute Angled Triangle
- Right-Angled Triangle
- Obtuse Angled Triangle

Let us discuss each type in detail.

#### Acute Angle Triangle

A triangle that has* all three angles less than 90*° is an acute angle triangle.

- So, all the angles of an acute angle triangle are called acute angles

Given below is an example of an acute angle triangle.

#### Right-Angle Triangle

A triangle that has * one angle that measures exactly 90*° is a right-angle triangle.

- The other two angles of a right-angle triangle are acute angles.
- The side opposite to the right angle is the largest side of the triangle and is called the hypotenuse.

In a right-angled triangle, the sum of squares of the perpendicular sides is equal to the square of the hypotenuse.

For e.g. considering the above right-angled triangle ACB, we can say:

(AC)^2 + (CB)^2 = (AB)^2

This is known as __Pythagoras theorem__

Vice versa, we can say that if a triangle satisfies the Pythagoras condition, then it is a right-angled triangle.

#### Obtuse/Oblique Angle Triangle

A triangle that has * one angle that measures more than 90*° is an obtuse angle triangle.

Given below is an example of an obtuse/oblique angle triangle.

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### Classification of triangles by length of sides

Based on the length of the sides, triangles are classified into three types:

- Scalene Triangle
- Isosceles Triangle
- Equilateral Triangle

Let us discuss each type in detail.

#### Scalene triangle

A triangle that has * all three sides of different lengths* is a scalene triangle.

- Since all the three sides are of different lengths, the
__three angles will also be different.__

Given below is an example of a scalene triangle

#### Isosceles triangle

A triangle that has * two sides of the same length and the third side of a different length* is an isosceles triangle.

- The
__angles opposite the equal sides measure the same.__

Given below is an example of an isosceles triangle.

#### Equilateral triangle

A triangle which has * all the three sides of the same length* is an equilateral triangle.

- Since all the three sides are of the same length, all
__the three angles will also be equal.__ - Each interior angle of an equilateral triangle = 60°

## Special cases of Right Angle Triangles

Let’s also see a few special cases of a right-angled triangle

### 45-45-90 triangle

In this triangle,

- Two angles measure 45°, and the third angle is a right angle.
- The sides of this triangle will be in the ratio – 1: 1: √2 respectively.
- This is also called an
since two angles are equal.__isosceles right-angled triangle__

### 30-60-90 triangle

In this triangle,

- This is a right-angled triangle, since one angle = 90°
- The angles of this triangle are in the ratio – 1: 2: 3, and
- The sides
*opposite to these angles*will be in the ratio – 1: √3: 2 respectively - This is a
since all three angles are different.__scalene right-angled triangle__

## The formula for Area of Triangle

- Area of any triangle = ½ * base * height
- Area of a right-angled triangle = ½ * product of the two perpendicular sides

## Properties of Triangle: Summary & Key Takeaways

Let us summarize some of the important properties of a triangle.

- The sum of all interior angles of any triangle is equal to
°__180__ - The sum of all exterior angles of any triangle is equal to
°__360__ - An exterior angle of a triangle is equal to the sum of its two interior opposite angles
- The sum of the lengths of
two sides of a triangle is always greater than the length of the third side__any__ - Similarly, the difference between the lengths of
__any__two sides of a triangle is always less than the length of the third side - The side opposite to the smallest interior angle is the shortest side and vice versa.
- Similarly, the side opposite to the largest interior angle is the longest side and vice versa.
- In the case of a right-angled triangle, this side is called the
*hypotenuse*

- In the case of a right-angled triangle, this side is called the
- The height of a triangle is equal to the length of the perpendicular dropped from a vertex to its opposite side, and this side is considered the base

If you liked this article, you may also like to read the following advanced level articles on triangles

- GMAT Geometry Concepts and Formulas on Triangles (Part-1)
- Properties of Triangles: Practice Questions (Part-2)
- Special Properties of Triangles (Part-3)

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## Properties of Triangle: Application quiz

## Question: 1

In an isosceles triangle DEF, if an interior angle ∠D = 100° then what is the value of ∠F?

- 20°
- 40°
- 60°
- 80°
- 100°

### Solution

**Step 1: Given**

- ∆DEF is an isosceles triangle
- ∠D = 100°

**Step 2: To find**

- The value of ∠F

**Step 3: Approach and Working out**

- We know that the sum of all interior angles in a triangle = 180°
- Implies, ∠D + ∠E + ∠F = 180°
- ∠E + ∠F = 180
^{0}– 100^{0}= 80° - Since ∆DEF is an isosceles triangle; two of its angles must be equal.
- And the only possibility is ∠E = ∠F
- Therefore, 2∠F = 80°
- Implies, ∠F = 40°

Hence the correct answer is **Option B.**

## Question 2

In a right-angled triangle, ∆ABC, BC = 26 units and AB = 10 units. If BC is the longest side of the triangle, then what is the area of ∆ABC?

- 120
- 130
- 240
- 260
- 312

### Solution

**Step 1: Given**

- ∆ABC is a right-angled triangle
- BC = 26 units
- AB = 10 units
- BC is the longest side of the triangle

**Step 2: To find**

- The area of triangle ∆ABC

**Step 3: Approach and Working out**

- We are given that BC is the longest side of the triangle, which implies that BC is the hypotenuse

Thus, according to Pythagoras rule:

- BC
^{2}= AB^{2}+ AC^{2} - 26
^{2}= 10^{2}+ AC^{2} - AC
^{2}= 676 – 100 = 576 - Therefore, AC = 24 units
- We know that the area of a right-angled triangle = ½ * product of the two perpendicular sides = ½ * AB * AC = ½ * 10 * 24 = 120 sq. units

Hence the correct answer is **Option A**.

**Here are a few more articles that you may like to read:**

- Properties of Quadrilateral
- Properties of Numbers: Even/Odd, Prime, and HCF & LCM
- Properties of Circle
- Properties of Lines and Angles

## FAQ – Properties of a triangle

**What is a triangle and its properties?**

A triangle is a closed figure with three sides, three vertices, three angles, and the sum of internal angles is 180°

**What are the different types of triangles?**

Triangles can be classified in 2 ways, according to internal angles and according to length of the sides. According to internal angles, there are three types of triangles i.e., acute, right, and obtuse-angled triangle. According to length of sides, triangles can be classified into 3 categories i.e., Scalene, Isosceles, and Equilateral triangle.

**What is a Scalene triangle?**

A triangle that has all three sides of different lengths is a scalene triangle.

**What is an Isosceles triangle?**

A triangle that has two sides of the same length and the third side of a different length is an isosceles triangle.

**What is an equilateral triangle?**

A triangle which has all the three sides of the same length is an equilateral triangle.

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