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A closed figure made with 3 line segments forms the shape of a triangle.

Triangles are an important part of our lives.

You can observe the shape of a triangle in signboards, your favourite sandwiches or birthday banners.

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A triangle is a closed figure or shape with 3 sides, 3 angles and 3 vertices.

A triangle with vertices P, Q, and R is denoted as △PQR.

The elements of a triangle are its 3 angles and its 3 sides.

Let's look at triangle PQR.

 

The elements of this triangle are:

• \(\angle PQR\)
• \(\angle QRP\)
• \(\angle RPQ\)
• PQ
• QR
• RP

You can explore the relation between the elements of a triangle by dragging the vertices of the triangle shown below.

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Triangles can be classified based on their angles or the length of their sides.

Classification of Triangles by Angles

Type of Triangle	Example	Explanation
Right-angled triangle		A right triangle has one of its vertex angles as 90°. In the above triangle, \(\angle \text{ACB}\) = 90°.
Acute-angled triangle		An acute triangle has all its internal angles as acute( i.e.. less than 90°).
Obtuse-angled triangle		An obtuse triangle has one of its internal angles as obtuse ( i.e.. greater than 90°).  In the given triangle, \(\angle \text{DFE}\)= 120\(^\circ\)

Classification of Triangles by Sides

Type of Triangle	Example	Explanation
Equilateral triangle		If all the three sides of a triangle are equal, it is called an equilateral triangle. In the given triangle, all three sides are equal in length. AB = BC = AC
Isosceles triangle		If any two sides of a triangle are equal, it is called an isosceles triangle. In the given triangle, two sides are equal. RS = RT
Scalene triangle		If a triangle has three unequal sides, it is called a scalene triangle. In the given triangle, all three sides are of unequal lengths. XZ ≠ XZ ≠ YZ

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• A triangle can always be divided into two right triangles, irrespective of its orientation.
• Triangles are polygons with the least number of sides. There are no polygons with less than 3 sides.
• Many towers, buildings and monuments are constructed in the shape of triangles to give them more stability and strength.
  Here are few examples for you:

1. Eiffel Tower

The world-famous Eiffel Tower is triangular in shape and has around 186 triangles in its structure

2. Egyptian Pyramids

They have triangular faces

3. Bridges

They use triangular frames for stability

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Some of the important properties of a triangle are listed below.

Property 1 - Angle Sum Property

The Angle Sum Property states that the sum of the three interior angles of a triangle is 180°

In the given triangle,

\(\angle \text{D} + \angle \text{E} + \angle \text{F}  = 180^\circ \)

Property 2 - Triangle Inequality Property

The Triangle Inequality Property sum of the length of the two sides of a triangle is greater than the third side.

in \(\Delta PQR \)

• p + q > r  ( 6 + 4 > 3)
• r + p > q (3 + 4  > 6)
• r + q > p ( 6 + 3 > 4)

Property 3 - Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is the sum of the squares of the other two sides.

\(\begin{align}\text{Hypotenuse}^2 &\!=\!\text{Base}^2 \!+\!\text{Altitude}^2 \end{align}\)

Property 4 - Side opposite the greater angle is the longest side

To understand the side opposite the greater angle is the longest side property, let's consider the below-given triangle.

In this triangle, \(\angle \text{A}\) is the greatest angle.

Thus, side BC is the longest side.

Property 5 - Exterior Angle Property

The Exterior Angle Property of a triangle is always equal to the sum of the interior opposite angles. 

In the given triangle, Exterior angle E1  = \(\angle \text{ABC} + \angle \text{BCA}\)

A triangle has 3 exterior angles and these exterior angles add up to 360° for any polygon.

Property 6 - Congruence Property

In the Congruence Property, two triangles are said to be congruent if all their corresponding sides and angles are congruent.

All  the corresponding angles and sides are equal,

• \(\angle\text{ ABC} = \angle\text{PQR}\)
• \(\angle\text{ BAC} = \angle\text{QPR}\)
• \(\angle\text{ BCA} = \angle\text{QRP}\)
• AB = PQ
• AC = PR
• BC = QR 

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The formulas of a triangle help us determine its area and perimeter.

Perimeter of a Triangle

Consider △XYZ.

The length of the sides of the triangle are represented as ‘a’, ‘b’ and ‘c’ respectively.

 

\(\text{Perimeter of Triangle} = a + b + c\)

Area of a Triangle

In the given triangle \(ABC\),

\(\begin{align}\text{Area of }\Delta ABC  = \frac{1}{2} h\times \text{BC}\end{align}\)

\(\text{BC}\) is the base and \(h\) is the height of the triangle.

\(\begin{align}\frac{1}{2} \times \text{Base} \times \text{Height}\end{align}\)