**Types of Triangles**** :**

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

There are six different types of triangles.

Here we are going to see, how triangles in geometry can be classified.

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

Each angle will be 60°.

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

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

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

And also all the three angles will be different.

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

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

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

**Problem 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°

From the above two points,

The given triangle is a scalene and obtuse triangle.

**Problem 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°

From the above two points, we have

The given triangle is a scalene and acute triangle.

**Problem 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°

From the above two points, we have

The given triangle is a scalene and right triangle.

**Problem 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°

From the above two points, we have

The given triangle is an isosceles and right triangle.

**Problem 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°

From the above two points, we have

The given triangle is an isosceles and acute triangle.

**Problem 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°

From the above two points, we have

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.

From the above point, we have

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.

From the above point, we have

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°

Hence, 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°

Hence, 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°

Hence, 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°

From the above two points, we have

The given triangle is an isosceles and right triangle.

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