# CLASSIFYING TRIANGLES

Classifying Triangles :

In geometry, triangles can be classified using various properties related to their angles and sides.

There are six different types of triangles.

1. Equilateral triangle

2. Isosceles triangle

3. Scalene triangle

4. Right triangle

5. Acute triangle

6. Obtuse triangle

Let us see about each type of triangles in detail.

## 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 will be 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 are obtuse (greater than 90 degree).

## Classifying Triangles - Examples

Example 1 :

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

Solution :

Let us consider the following two important points related to the given information.

(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.

Example 2 :

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

Solution :

Let us consider the following two important points related to the given information.

(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.

Example 3 :

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

Solution :

Let us consider the following two important points related to the given information.

(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.

Example 4 :

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

Solution :

Let us consider the following two important points related to the given information.

(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.

Example 5 :

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

Solution :

Let us consider the following two important points related to the given information.

(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.

Example 6 :

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

Solution :

Let us consider the following two important points related to the given information.

(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.

Example 7 :

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

Solution:

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

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

Then,

(3x + 3)° < 90°

3x + 3 < 90

Subtract 3 from each side.

3x < 87

Divide each side by 3.

x < 29

So, the value of x is less than 29.

Example 8 :

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

Solution:

Because 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 + 4  =  90

Subtract 4 from each side.

2x  =  86

Divide each side by 2.

x  =  43

So, the value of x is 43.

Example 9 :

If (2x)°, y° and (3z)° are the angles of a acute triangle, then find the value of z.

Solution:

Because 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°

3z < 90

Divide each side by 3.

z < 30

So, the value of z is less than 30.

Example 10 :

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

Solution:

The angle sum property of a triangle states that the angles of a triangle always add up to 180°.

Then,

(2x + 15)° + (3x)° + (6x)°  =  180°

2x + 15 + 3x + 6x  =  180

Simplify.

11x + 15  =  180

Subtract 15 from each side.

11x  =  165

Divide each side by 11.

x  =  15

Substitute 15 for x into the given expressions to find the angles of the triangle.

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. After having gone through the stuff given above, we hope that the students would have understood how to classify triangles.

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