TYPES OF TRIANGLES

Triangles can be broadly classified into two types. They are 

• Triangles based on the lengths of their sides.
• Triangle based on their angles.

Equilateral Triangle

An equilateral triangle is a triangle in which all the three sides will be equal.

Each angle will be 60°.

Isosceles Triangle

A triangle with two equal sides is called as isosceles triangle.

The angles corresponding to the equal sides will always be equal.

Scalene Triangle

In a scalene triangle the length of all the three sides will be different.

And also all the three angles will be different. 

Right Triangle

A right triangle is the triangle in which one of the angles is 90°.

Acute Triangle

An acute triangle is a triangle with all three angles are less than 90 degree.

Obtuse Triangle

An obtuse triangle is a triangle in which one of the angles is obtuse (greater than 90 degree).

Practice Problems

Problem 1 :

Identify the type of triangle whose angles are 35°, 40°, 105°.

Solution :

(i)  All the given three angles are different.

(ii)  One of the angles is greater than 90°

So, the given triangle is a scalene and obtuse triangle. 

Problem 2 :

Identify the type of triangle whose angles are 55°, 65°, 60°.

Solution :

(i)  All the given three angles are different.

(ii)  All the three angles are less than 90°

So, the given triangle is a scalene and acute triangle. 

Problem 3 :

Identify the type of triangle whose angles are 50°, 40°, 90°.

Solution :

(i)  All the given three angles are different.

(ii)  One of the angles is 90°

So, the given triangle is a scalene and right triangle. 

Problem 4 :

Identify the type of triangle whose angles are 45°, 45°, 90°.

Solution :

(i)  Two of the given angles are equal

(ii)  One of the angles is 90°

So, the given triangle is an isosceles and right triangle. 

Problem 5 :

Identify the type of triangle whose angles are 70°, 70°, 40°.

Solution :

(i)  Two of the given angles are equal

(ii)  All the three angles are less than 90°

So, the given triangle is an isosceles and acute triangle. 

Problem 6 :

Identify the type of triangle whose angles are 30°, 30°, 120°.

Solution :

(i)  Two of the given angles are equal

(ii)  One of the angles is greater than 90°

So, the given triangle is an isosceles and obtuse triangle. 

Problem 7 :

Identify the type of triangle whose sides are 5 cm, 6 cm and 7 cm. 

Solution :

The length of all the three sides are different. 

So, the given triangle is a scalene triangle.

Problem 8 :

Identify the type of triangle whose sides are 6 cm, 6 cm and 8 cm. 

Solution :

The lengths of two of the sides are equal.

So, the given triangle is an isosceles triangle.

Problem 9 :

If (3x + 3) is one of the angles of an acute triangle, then find the value of "x". 

Solution :

Since the given triangle is acute triangle, all the three angles will be less than 90°. 

So, (3x + 3) will also be less than 90°.

Then, 

3x + 3  < 90°

3x < 87°

x < 29°

So, the value of "x" is less than 29°.

Problem 10:

If 50°, 40° and (2x + 4)°are the angles of a right triangle, then find the value of "x". 

Solution:

Since the given triangle is a right triangle, one of the angles must be 90°

In the given three angles 50°, 40° and (2x+4)°, the first two angles are not right angles. 

So the third angle (2x+4)° must be right angle. 

Then,

2x + 4  =  90°

2x  =  86°

x  =  43°

So, the value of "x" is  43°.

Problem 11 :

If 2x, y and 3z are the angles of a acute triangle, find the value of "z". 

Solution:

Since the given triangle is acute triangle, all the three angles will be less than 90°. 

So, 3z will also be less than 90°.

Then, 

3z  <  90°

z  <  30°

So, the value of "z" is less than 30°.

Problem 12 :

If 2x+15, 3x and 6x are the angles of a triangle, identify type of the triangle. 

Solution :

Since 2x+15, 3x and 6x are the angles of a triangle, by property

(2x + 15) + 3x + 6x  =  180°

11x + 15  =  180°

11x  =  165°

x  =  15°

Plugging x = 15°, we get

First angle  =  2x + 15  =  2(15) + 15  =  45°

Second angle  =  3x  =  3(15)  =  45°

Third angle  =  6x  =  6(15)  =  90°

Let us consider the following two important points from the above calculation. 

(i)  Two of the angles are equal

(ii)  One of the angles is 90°

So, the given triangle is an isosceles and right triangle. 

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