Triangle definition, types and properties

In this post we will discuss about Triangle basic definitions, its type and property of triangle.

We have tried to explain the concept in as simple way as possible but even if you face any doubt feel free to ask in the comment section.

Basic concepts of Triangle

What is a Triangle ?

If the polygon is made of three sides, then it is called a triangle.

This polygon consists of three sides and hence, called a triangle and is written as  triangle ABC

Note – A triangle is the smallest polygon with three sides

Classification of Triangle

On the basis of length of sides
a. Scalene Triangle
b. Isosceles Triangle
c. Equilateral Triangle

On the basis of Angles
a. Acute angle Triangle
b. Obtuse angle Triangle
c. Right angle triangle

Scalene Triangle

If all the sides of the triangle are of different lengths, then the triangle is called scalene triangle. (i.e. no sides of the triangle are congruent).

Here, AB ≠ BC ≠ CA
Hence, it is a scalene triangle

Isosceles Triangle

If only two sides of the triangle are of equal lengths, then the triangle is called isosceles triangle (i.e. two sides of the triangle are congruent)

Here, DE = EF
Hence, it is an isosceles triangle

Isosceles triangle theorem

For an isosceles triangle,
If two sides of a triangle are equal (congruent), then the angles opposite to these sides are also equal (congruent)
Or
Conversely, if two angles of a triangle are equal (congruent), then the sides opposite of those angles are also equal (congruent)

Here, AB = AC
Hence, it is an isosceles triangle
Therefore, ∠ABC = ∠ACB

Equilateral Triangle

If all the sides of the triangle are of equal lengths, then the triangle is called equilateral triangle (i.e. all the sides of the triangle are congruent)

Note – For an equilateral triangle, all the 3 interior angles are equal to 60^o
∠A = ∠B = ∠C = 180^o

Acute angled triangle

If all the 3 interior angles of a triangle are acute (less than 90^o), then the triangle is called acute angled triangle

Here, in both the above figures, all the 3 interior angles are less than 90o hence are acute angled triangle

Obtuse angled triangle

If there is one interior angle obtuse (greater than 90^o) in the triangle, then the triangle is called obtuse angled triangle

Here, angle > 90^o
Hence, obtuse angled triangle

Right angled triangle

If there is one interior angle equal to 90^o in a triangle, then the triangle is called right angled triangle.

Here, angle = 90^o
Hence, right angled triangle

For a right-angled triangle, the side opposite to the 90^o angle is called “Hypotenuse” i.e. the longest side, the other two sides are called “Perpendicular” and “Base”.

What is Pythagoras theorem of Right angled Triangle?

Pythagoras theorem states that “In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides”. The sides of this triangle have been named as Perpendicular, Base and Hypotenuse and “Base”.

Question asked from Pythagoras theorem

If the length of perpendicular is 8 cm and base is 6 cm of a right-angled triangle as shown in the figure. Find the length of the longest side (Hypotenuse) of that triangle.

Sol. Let hypotenuse = x cm

By the theorem we know;

Hypotenuse² = Base² + Perpendicular²
==> x² = 6² + 8²
==> x² = 36 + 64 = 100
==> x = √100 = 10
Therefore, we found the value of hypotenuse = 10 cm

On the basis of both side and angles there are two more kinds of triangles –

Isosceles right triangle

Triangle with two sides of equal length and one right angle is called isosceles right triangle.
In this type of triangle, each of the other two angles are equal to 45^o

Ex –

Here, BC = AC and also ∠ACB = 90^o
Also, ∠ABC = ∠ACB = 45^o
Hence, it is an isosceles right triangle

Scalene obtuse triangle

Triangle with all the sides of different length and one obtuse interior angle, is called scalene obtuse triangle.

Here, angle > 90^o & all sides are of different length
Hence, scalene obtuse triangle

Congruent Triangle

Congruent triangles have exactly the same shape and size i.e. if two triangles have exactly same size and same shape then they are said to be congruent of each other

Both the above triangles are of same size and shape. Hence both are congruent triangles

How to construct a congruent triangle of a given triangle

To construct:   XYZ congruent to   ABC

Step 1: Draw a line l and with the help of the compass measure AB and on line from point X cut y such that XY = AB

Step 2: Taking X as center cut an arc for Z with the help of compass by opening the compass to the length of AC. Similarly, take Y as center and cut an arc on previous arc by opening the compass to the length of BC

Step 3: The intersection of both the arc is point Z. Marl it as Z and then join all the three points X, Y and Z.
The triangle formed is  XYZ congruent to ABC.