# Triangle and its Types

## Introduction

We have already discussed about polygons in the Polygon and its Types chapter. There we learnt that a polygon is a closed curve, which is made up of only line segments.

Triangle is one type of polygon which has three sides or line segments. It is a polygon with least number of sides, in other words, a polygon with less than three sides does not exist. Triangle is a closed figure which is formed by joining three line segments. It has 3 sides, 3 vertices and 3 angles. Triangle is denoted by Δ, a delta symbol.

Lets learn about triangles with the following figure with a triangle whose name is written as ΔABC, where Δ is a symbol of triangle and ABC is the name of a triangle, which always includes three vertices of a triangle.

Example of a triangle ΔABC

We can use any three vertices in any order to represent a triangle.
For example, ABC or BAC or CAB or ACB. Therefore, the triangle can also be written as ΔBAC or ΔCAB or ΔACB.

## Sides of triangle

As you know that triangle always has three sides. These sides are also called the line segments.
So, in ΔABC, line segments AB, BC and CA are sides of the ΔABC. These three sides of a triangle are written as $\stackrel{—}{\mathrm{AB}}$, $\stackrel{—}{\mathrm{BC}}$ and $\stackrel{—}{\mathrm{CA}}$.

## Vertex of triangle

Vertex of a triangle is that point where any two sides of a triangle meet out or intersect.
In the ΔABC in the above figure, sides AB and AC meet at point A. So, A is the vertex of ΔABC.
Similarly, sides BC and AB meet at point B. So B is a vertex of ΔABC.
Also, sides AC and BC meet at point C. So C is a vertex of ΔABC.
Therefore, A, B and C are the three vertices of ΔABC.

## Angles of triangle

Angles are formed at the vertex of a triangle. It is a measurement of how slanted two lines are to each other. The angles are measured in degrees units.
Let’s understand angles in a triangle ΔABC from the above figure.
As we already discussed above, ΔABC has three vertices A, B and C.

Therefore, we can say angles are formed at vertices A, B and C. Angles are written as ∠ABC, ∠BAC and ∠ACB or in a short form as ∠B, ∠A and ∠C respectively.
In other words, we can say:
∠ABC or ∠B is formed at vertex B of ΔABC.
∠BAC or ∠A is formed at vertex A of ΔABC.
∠ACB or ∠C is formed at vertex C of ΔABC.

## Types of triangle on the basis of length of sides

Triangle has many types depending upon the length of its sides.

### Equilateral triangle

A triangle is said to be an equilateral triangle if all sides of a triangle are of equal length.
ΔABC is equilateral triangle because AB = BC = CA, where AB is length of side AB, BC is length of side BC and CA is the length of side CA.

Equilateral triangle ΔABC

### Isosceles triangle

A triangle is said to be an Isosceles triangle if any two sides of the triangle are of equal length.
ΔABC is Isosceles triangle because AB = AC.

Isosceles triangle ΔABC

### Scalene triangle

A triangle is said to be a scalene triangle if all sides of the triangle are of unequal length.
ΔABC is Scalene triangle because AB ≠ BC ≠ AC.

Scalene triangle ΔABC

## Types of triangle on the basis of size of angles

Triangle has many types depending upon the length of its angles.

### Acute angled triangle

A triangle is said to be an acute angled triangle if each angle of a triangle is acute. Acute angle is that angle which is less than 90°.
So, the ΔABC in figure is an acute angled triangle because all angles are less than 90°. That is, ∠A, ∠B and ∠C are all less than 90°.

Acute angled triangle ΔABC

### Right angled triangle

A triangle is said to be a right angle triangle if one of its angles is right angle. Right angle is that angle whose measure is 90°.
So, the ΔABC in figure is a right angled triangle because ∠C is equal to 90°.

Right angled triangle ΔABC

### Obtuse angled triangle

A triangle is said to be an obtuse angled triangle if one of its angles is an obtuse angle. Obtuse angle is that angle which is greater than 90°.
So, the ΔABC in figure is an obtuse angled triangle because ∠C is greater than 90°.

Obtuse angled triangle ΔABC

## List of types of triangles on the basis of sides

Name of triangleNumber of equal sides
Equilateral triangleAll 3 sides are equal
Isosceles triangleAny 2 sides are equal
Scalene triangleNo sides are equal

## List of types of triangles on the basis of angle

Name of triangleMeasure of angle
Acute angled triangleAll three angles are acute angles
Right angled triangleOne angle is 90°
Obtuse angled triangleOne angle is obtuse angle

### 1) What is a triangle?

Triangle is a plane and closed figure which is formed by three line segments.

### 2) How many sides are there in a triangle?

There are 3 sides in a triangle.

### 3) How many angles are there in a triangle?

There are 3 angles in a triangle.

### 4) What are the types of a triangle according to its sides?

There are three types of a triangle according to its sides.
1. Equilateral triangle
2. Isosceles triangle
3. Scalene triangle

### 5) What are the types of a triangle according to its angles?

There are three types of a triangle according to its angles.
1. Right angled triangle
2. Obtuse angle triangle
3. Acute angle triangle

### 6) What is the sum of three angles of a triangle?

Sum of three angles of a triangle is equal to 1800.

## Solved Examples

### 1) In isosceles triangle ΔABC, with AB=AC, if base angle = 300, find the other angles.

Solution

In isosceles ΔABC

AB = AC

∴ ∠B = ∠C ( ∵ base angles are equal )

∠C = 300

In ΔABC,

∠A + ∠B + ∠C = 1800

∠A + 300 + 300 = 1800

∠A + 600 = 1800

∠A = 1800 - 600

∠A = 1200

### 2) In right angled triangle ΔABC at B, if ∠C = 300, find ∠A.

Solution

In right angled triangle ΔABC

∠A + ∠B + ∠C = 1800

∠A + 900 + 300 = 1800

∠A + 1200 = 1800

∠A = 1800- 1200

∠A = 600

### 3) In triangle ΔABC, if ∠A = 500, ∠B = 300, find ∠C.

Solution

In triangle ΔABC

∠A + ∠B + ∠C = 1800

500 + 600 + ∠C= 1800

1100 + ∠C= 1800

∠C = 1800- 1100

∠C = 700

### 4) If the angles of ΔABC are in ratio 1 : 2 : 3, find all angles.

Solution

Let the angles be x, 2x and 3x

In triangle, sum of three angles is 1800.

∴ ∠A + ∠B + ∠C = 1800

x + 2x + 3x = 1800

6x = 1800

x = $\frac{180}{6}$

x = 300

∴ the three angles are:

x = 300

2x = 2 x 300 = 600

3x = 3 x 300 = 900

∴ the three angles are 300, 600 and 900

## Multiple Choice Questions

a) x

b) y

c) z

d) ∠A

a) 1800

b) 3600

c) 7200

d) 14400

a) 500

b) 400

c) 600

d) 700

a) 300

b) 400

c) 900

d) 600

### 5) A triangle whose all sides are equal is called as

a) scalene triangle

b) acute triangle

c) equilateral triangle

d) isosceles triangle

a) 900

b) 1000

c) 1100

d) 1200

a) true

b) false

c) maybe

d) none of these

### 8) A triangle whose two sides are equal is called as

a) scalene triangle

b) equilateral triangle

c) isosceles triangle

d) right angled triangle

### 9) A triangle with one angle more than 900 is called as

a) scalene triangle

b) obtuse triangle

c) equilateral triangle

d) right angled triangle

### 10) A triangle with one angle less than 900 is called

a) obtuse triangle

b) acute triangle

c) right angled triangle

d) equilateral triangle