A triangle is a simple closed figure made of three-line segments. We come across many different shapes in triangles. Classification of triangles is done based on the length of their sides and the measure of their angles.

What is a Triangle?

A plane figure formed by three non-parallel line segments is called a triangle.

Parts of a Triangle

A triangle has nine parts or elements, namely three sides, three angles and three vertices.

If $P,Q,R$ are three non-collinear points on the plane of the paper, then the figure made up by the three-line segments $PQ,QR$ and $RP$ is called a triangle with vertices $P,Q$ and $R$.

The triangle with vertices $P,Q$ and $R$ is generally denoted by the symbol $\mathrm{\Delta }PQR$.

Note that the triangle $\mathrm{\Delta }PQR$ consists of all the points on the line segments $PQ,QR$ and $RP$.

Sides: The three-line segments $PQ,QR$ and $RP$, that form the triangle $\mathrm{\Delta }PQR$ are called the sides of the triangle $\mathrm{\Delta }PQR$.

Angles: The three angles $\mathrm{\angle }QPR,\mathrm{\angle }PQR$ and $\mathrm{\angle }PRQ$ are called the angles of $\mathrm{\Delta }PQR$.

For the sake of convenience, we shall denote angles $\mathrm{\angle }QPR,\mathrm{\angle }PQR$ and $\mathrm{\Delta }PQR$ by angles $\mathrm{\angle }P,\mathrm{\angle }Q$ and $\mathrm{\angle }R,$ respectively.

Vertices: The vertex is the point in a triangle where the two sides meet. In the $\mathrm{\Delta }PQR$, the points $P,Q$ and $R$ are the vertices.

Elements or parts: The three sides $PQ,QR,RP,$ angles $\mathrm{\angle }P,\mathrm{\angle }Q,\mathrm{\angle }R$ of a triangle $PQR$ are together called the six parts or elements of the triangle $PQR$.

Classification of Triangles

Triangles are classified based on $(a)$ the length of sides $(b)$ based on the measure of the angles.

Classification of Triangles based on the Length of Sides

Scalene Triangle

A triangle having three sides of different lengths is called a scalene triangle.

Here, in scalene $\mathrm{\Delta }ABC$, the sides $AB,BC$ and $AC$ are of different measures.

Isosceles Triangle

A triangle having two sides equal is called an isosceles triangle.

Here in isosceles $\mathrm{\Delta }PQR$, sides $PQ$ and $PR$ are two equal sides, and $\mathrm{\angle }PQR$ and $\mathrm{\angle }PRQ$ are two equal angles.

Equilateral Triangle

A triangle having all the sides equal is called an equilateral triangle”.

Note that an equilateral triangle is an acute-angled triangle because the measure of its every angle is ${60}^{\circ }$.

$\mathrm{\Delta }DEF$ is an equilateral triangle, sides $DE,EF$ and $DF$ are equal in measure, and angles $\mathrm{\angle }DEF,\mathrm{\angle }DFE$ and $\mathrm{\angle }EDF$ measure the same ${60}^{\circ }$.

Some important facts:

1. The angles opposite to equal sides of an isosceles triangle are equal.
2. A scalene triangle has no two angles equal.

Classification of Triangles based on the Measure of Angles

Acute Angle Triangle

A triangle each of whose angles measures less than ${90}^{\circ }$ is called an acute-angled triangle or simply an acute triangle.

$\mathrm{\Delta }XYZ$ is an acute-angled triangle, here $\mathrm{\angle }YXZ={60}^{\circ },\mathrm{\angle }XYZ={70}^{\circ }$ and $\mathrm{\angle }XZY={50}^{\circ }.$

Right Angle Triangle

A triangle whose one angle is a right angle is called a right-angled triangle or a right triangle.

The side opposite to the right angle in a right-angled triangle is known as the hypotenuse of the triangle, and the other two sides are called the legs of the triangle.

In a right-angled triangle, one angle is ${90}^{\circ }$, and the remaining two angles of a right-angled triangle are acute.

$\mathrm{\Delta }DEF$ is a right-angled triangle, right angle at $E$. The remaining two angles, $\mathrm{\angle }D$ and $\mathrm{\angle }F$ are acute angles, that is $\mathrm{\angle }D={50}^{\circ }$ and $\mathrm{\angle }F={40}^{\circ }$.

Right Isosceles Triangle

In a right-angled triangle, if the measure of each acute angle is equal to ${45}^{\circ }$, then it is called a right isosceles triangle.

Obtuse Angle Triangle

A triangle one of whose angles measures more than ${90}^{\circ }$ is called an obtuse-angled triangle or simply an obtuse triangle.

Note that only one angle can be obtuse in a triangle, and the remaining two angles are acute angles.

$\mathrm{\Delta }PQR$ is an obtuse-angled triangle. Here, $\mathrm{\angle }PQR={120}^{\circ }$ (greater than ${90}^{\circ }$) and the remaining two angles are acute angles; those are $\mathrm{\angle }QPR={15}^{\circ }$ and $\mathrm{\angle }PRQ={45}^{\circ }$.

Some important facts:

1. Each angle of an equilateral triangle measures ${60}^{\circ }$.
2. A triangle cannot have more than one right angle.
3. A triangle cannot have more than one obtuse angle.

Angle Sum Property of a Triangle

The sum of the interior angles of a triangle is ${180}^{\circ }$ or a measure of $2$ right angles.

Try these:

1. Draw some equilateral triangles and measure each one of the angles of each such triangle. You will find that the measure of each angle is ${60}^{\circ }$. Thus, the measure of each angle of an equilateral triangle is ${60}^{\circ }$.
2. Draw an isosceles triangle and measure its angles. You will find that the angles opposite to equal sides are equal. Thus, an isosceles triangle has exactly two equal angles.
3. Draw a scalene triangle, i.e. a triangle whose sides are of different lengths. Measure the angles of this triangle. You will find that the measures of the three angles are different. Thus, a scalene triangle has no two angles equal.

Solved Examples on Types of Triangles

Q.1. In an isosceles triangle, if the measure of each equal angle is ${50}^{\circ }$, then find the measure of the third angle.
Ans:
Given:
The measure of each equal angle in an isosceles triangle is ${50}^{\circ }$
Let the measure of the third angle be $x$.

We know that the sum of the measure of interior angles of a triangle is ${180}^{\circ }$.
Now, ${50}^{\circ }+{50}^{\circ }+x={180}^{\circ }$
$\Rightarrow x={180}^{\circ }–100={80}^{\circ }$
Hence, the measure of the third angle is ${80}^{\circ }$.

Q.2. If the perimeter of an equilateral triangle is $18\mathrm{c}\mathrm{m},$ then find the measure of each side.
Ans:
Given: 
The perimeter of an equilateral triangle is $18\mathrm{c}\mathrm{m},$
As we know, the measure of each angle in an equilateral triangle is the same.
Let the measure of each side of the equilateral triangle be $x\mathrm{c}\mathrm{m}.$
We know that the perimeter of any polygon is the total length of the boundary of a polygon.
Then, $x+x+x=18\mathrm{c}\mathrm{m}$
$\Rightarrow 3x=18\mathrm{c}\mathrm{m}$
$\Rightarrow x=6\mathrm{c}\mathrm{m}$
Hence, the measure of each side of the equilateral triangle is $6\mathrm{c}\mathrm{m}.$

Q.3. One of the acute angles of a right-angle triangle is ${50}^{\circ }$. Find the other acute angle.
Ans:
Given: 
One of the acute angles of a right-angle triangle is ${50}^{\circ }$.
Let the measure of the other acute angle be $x$.
We know that the measure of one of the angles in the right-angled triangle is ${90}^{\circ }$.

Now, ${50}^{\circ }+{90}^{\circ }+x={180}^{\circ }$ (the sum of the measure of interior angles of a triangle is ${180}^{\circ }$)
$\Rightarrow {50}^{\circ }+{90}^{\circ }+x={180}^{\circ }$
$\Rightarrow {140}^{\circ }+x={180}^{\circ }$
$\Rightarrow x={180}^{\circ }–{140}^{\circ }={40}^{\circ }$
Hence, the measure of the other acute angle is ${40}^{\circ }$

Q.4. If one angle is twice the smallest angle and another angle is three times the smallest angle, then find the measure of each angle.
Ans:
Let the measure of the smallest angle be $x$.
Then, one angle is twice the smallest angle $=2x$.
Another angle is three times the smallest angle $=3x$.

Now, $x+2x+3x={180}^{\circ }$ (the sum of the measure of interior angles of a triangle is ${180}^{\circ }$)
$\Rightarrow 6x={180}^{\circ }$
$\Rightarrow x={30}^{\circ }$
Then, $2x={60}^{\circ }$ and $3x={90}^{\circ }$
Hence, the measure of all three angles is ${30}^{\circ },{60}^{\circ }$ and ${90}^{\circ }$.

Q.5. If the measure of one of the angles is ${140}^{\circ }$ and the other two angles are equal, then find the measure of equal angles.
Ans:
Given:
The measure of one of the angles is ${140}^{\circ }$
Let the measure of each equal angles be $x$.

Now, ${140}^{\circ }+x+x={180}^{\circ }$
$\Rightarrow 2x={180}^{\circ }–{140}^{\circ }={40}^{\circ }$
$\Rightarrow x={20}^{\circ }$
Hence, the measure of each angle is ${20}^{\circ }$.

Summary

In this article, we learnt about different types of triangles such as Scalene Triangle, Isosceles Triangle, Equilateral Triangle, Acute Angle Triangle, Right Angle Triangle, Obtuse Angle Triangle. We have also seen how we can find any parameter when types of the triangle and some other parameter of the triangle are given. This plays an important role in geometry.

Frequently Asked Questions (FAQs) on Types of Triangles

Q.1. What are the 7 types of a triangle?
Ans: The seven types of triangles are equilateral triangle, scalene triangle, isosceles triangle, acute-angled triangle, right-angled triangle, right isosceles triangle, and obtuse-angled triangle.

Q.2. How many types of triangles are there, and their definition?
Ans: Classification of triangles is based on the measure of their sides and angles. Based on the measure of their sides and angles, six basic triangles are there:
Naming triangles by considering the lengths of their sides:
Scalene Triangle: A triangle having three sides of different lengths is called a scalene triangle.
Isosceles Triangle: A triangle having two sides equal is called an isosceles triangle.
Equilateral Triangle: A triangle having all sides equal is called an equilateral triangle.
Note that an equilateral triangle is an acute-angled triangle because the measure of its every angle is ${60}^{\circ }$.
Naming triangles by considering the measures of their angles:
Acute Triangle: A triangle each of whose angles measures less than ${90}^{\circ }$  is called an acute-angled triangle or simply an acute triangle.
Right Triangle: A triangle whose one angle is a right angle is called a right-angled triangle or a right triangle.
Obtuse Triangle: A triangle one of whose angles measures more than ${90}^{\circ }$ is called an obtuse-angled triangle or simply an obtuse triangle.

Q.3. What are the six types of a triangle?
Ans: Naming triangles by considering the lengths of their sides: Equilateral triangle, Scalene triangle, and Isosceles triangle.
Naming triangles by considering the measures of their angles: Acute angled triangle, right-angled triangle, right isosceles triangle, and obtuse-angled triangle.

Q.4. What are the 4 different types of triangles based on the measure of their angles?
Ans: Naming triangles by considering the measures of their angles: Acute angled triangle, right-angled triangle, right isosceles triangle, and obtuse-angled triangle.

Q.5. What are the 3 types of triangles based on the lengths of their sides?
Ans: Naming triangles by considering the lengths of their sides: Equilateral triangle, Scalene triangle, and Isosceles triangle.

Q.6. What is a triangle called?
Ans: A plane figure formed by three non-parallel line segments is called a triangle.

Related Concepts:

Area of Triangle Concepts

Area of Triangle in Coordinate Geometry

Area of Triangle & Properties

Properties of Triangle Concepts

Area of Triangle Vectors Cross Product Concepts

Important Points of Triangle

Examples on Properties of Triangle

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