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 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 *tri–angle* 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, and three vertices.

- And, the sum of all three angles of any triangle equals to 180
^{0}.

## Basic Properties of Triangle

Following are the basic properties of triangles:

- The sum of all internal angles of a triangle is always equal to 180
^{0. }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.

## Properties of Triangle: Classification of Triangles by Type

Based on the internal angles and the length of its sides, triangles can be classified in 2 major ways:

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

### Properties of Triangle: Classification of a triangle by internal angles

Based on the angle measurement, triangles are classified into three types:

- 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^{0}* 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^{0}* 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 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^{0}* is an obtuse angle triangle.

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

Questions on triangles are very commonly asked on the GMAT. Ace GMAT Quant by signing up for our free trial and get access to 400+ questions. We are the most reviewed company on gmatclub.

Learn from Guillermo who improved from Q38 to Q50.

### Properties of Triangle: 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
^{0}

## Properties of Triangle: 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
^{0}, 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
^{0} - 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 triangle.

- The sum of all interior angles of any triangle is equal to
.__180__^{0} - The sum of all exterior angles of any triangle is equal to
.__360__^{0} - 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 case of a right-angled triangle, this side is called the
*hypotenuse*

- In 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

## Properties of Triangle: Application quiz

## Question: 1

In an isosceles triangle DEF, if an interior angle ∠D = 100^{0} then what is the value of ∠F?

- 20
^{0} - 40
^{0} - 60
^{0} - 80
^{0} - 100
^{0}

### Solution

**Step 1: Given**

- ∆DEF is an isosceles triangle
- ∠D = 100
^{0}

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

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 Triangles

**What is a triangle?**

A triangle is closed figure with three sides and sum of internal angles as 180°

**How can we classify 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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