# Triangles

## 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?

A close figure having three side.

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:

1. A triangle has three sides, three angles, and three vertices.
2. The sum of all internal angles of a triangle is always equal to 180°. This is called the angle sum property of a triangle.
3. The sum of the length of any two sides of a triangle is greater than the length of the third side.
4. The side opposite to the largest angle of a triangle is the largest side.
5. 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 – 3
• Classification according to the length of its sides – 3

### Classification of a triangle by internal angles

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

1. Acute Angled Triangle
2. Right-Angled Triangle
3. 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

### 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.

## Classification of triangles by length of sides

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

1. Scalene Triangle
2. Isosceles Triangle
3. 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.

### 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.

### 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 isosceles right-angled triangle since two angles are equal.

#### 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 scalene right-angled triangle since all three angles are different.

#### 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

#### NCERT SOLUTIONS FOR CLASS 11 MATHS CHAPTER 7 / PERMUTATIONS AND COMBINATIONS EX 7.1 /

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