**Classifying triangles by angles :**

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

In this section, we are going to study how triangles can be classified by its angles.

**Note : **

1. In a triangle, if all the three angles or the lengths of all the three sides are equal, then it is equiangular or equilateral triangle.

2. In a triangle, if two of the angles are equal and the third angle is less than 90°, then it is isosceles and acute triangle.

3. In a triangle, if two of the angles are equal and the third angle is greater 90°, then it is isosceles and obtuse triangle.

4. In a triangle, if all the three angles are different and each of the three angles is less than 90°, then it is scalene and acute triangle.

5. In a triangle, if all the three angles are different and one of the angles is greater than 90°, then it is scalene and obtuse triangle.

**Example 1 :**

Identify the type of triangle whose diagram is given below.

**Solution :**

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

(i) All the given three angles are different.

(ii) Each of the three angles is less than 90°

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

**Example 2 :**

Identify the type of triangle whose diagram is given below.

**Solution :**

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

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

(ii) Two sides are equal in length. The angles formed by the two congruent sides with the third side also must be equal.

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

**Example 3 :**

Identify the type of triangle whose diagram is given below.

**Solution :**

In the triangle above, the lengths of all the three sides are same and all the three angles are congruent.

So, the given triangle is equiangular triangle.

**Example 4 :**

Identify the type of triangle whose diagram is given below.

**Solution :**

In the triangle above, two sides are congruent. The angles formed by the third side with two congruent sides will always be equal.

The diagram given below illustrates this.

By Triangle Sum theorem, we have

x + x + 40° = 180°

2x + 40° = 180°

Subtract 40° from both sides.

2x = 140°

Divide both sides by 2.

x = 70°

The angles measures of the given triangle are 40°, 70° and 70°.

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

(i) Two of the angles are equal

(ii) Each of the three angles is less than 90°.

From the above two points, we have

The given triangle is an isosceles and acute triangle.

**Example 5 :**

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

**Solution:**

Since 2x+15, 3x and 6x are the angles of a triangle, by Triangle Sum Theorem, we have

(2x+15) + 3x + 6x = 180°

11x + 15 = 180°

11x = 165°

x = 15°

Substitute x = 15° in the given angles measures 2x+15, 3x and 6x.

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.

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

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