**Sum of the Three Angles of a Triangle:**

In this section, we are going to see what the three angles of a triangle add up to.

In any triangle,

**Sum of the three angles = 180°**

More clearly,

If the sum of the three angles is not equal to 180°, then we can conclude that the three angles can not be the angles of a triangle.

**Example 1 :**

Can 30°, 60° and 90° be the angles of a triangle ?

**Solution :**

Let us add all the three given angles and check whether the sum is equal to 180°.

30° + 60° + 90° = 180°

Because the sum of the angles is equal 180°, the given three angles can be the angles of a triangle.

**Example 2 : **

Can 35°, 55° and 95° be the angles of a triangle ?

**Solution :**

Let us add all the three given angles and check whether the sum is equal to 180°.

35° + 55° + 95° = 185°

Since the sum of the angles is not equal 180°, the given three angles can not be the angles of a triangle.

**Example 3 : **

In a triangle, If the second angle is 5° greater than the first angle and the third angle is 5° greater than second angle, find the three angles of the triangle.

**Solution :**

Let "x" be the first angle.

Then, the second angle = x + 5

The third angle = x + 5 + 5 = x + 10

We know that,

the sum of the three angles of a triangle = 180°

x + (x + 5) + (x + 10) = 180°

3x + 15 = 180

3x = 165

x = 55

The first angle = 55°

The second angle = 55 + 5 = 60°

The third angle = 60 + 5 = 65°

Hence, the three angles of a triangle are 55°, 60° and 65°.

**Example 4 : **

If the angles of a triangle are in the ratio 2 : 7 : 11, then find the angles.

**Solution :**

The angles of the triangle are in the ratio 2 : 7 : 11.

Then, the three angles are

2x, 7x and 11x

In any triangle,

Sum of the three angles = 180°

So, we have

2x + 7x + 11x = 180°

20x = 180

x = 9

Then, the first angle = 2x = 2 ⋅ 9 = 18°

The second angle = 7x = 7 ⋅ 9 = 63°

The third angle = 11x = 11 ⋅ 9 = 99°

Hence the angles of the triangle are 18°, 63° and 99°.

**Example 5 : **

In a triangle, If the second angle is 20% more than the first angle and the third angle is 20% less than the first angle, then find the three angles of the triangle.

**Solution :**

Let "x" be the first angle.

Then, the second angle = 120% of x = 1.2x

The third angle = 80% of x = 0.8x

We know that,

the sum of the three angles of a triangle = 180°

x + 1.2x + 0.8x = 180°

3x = 180°

x = 60°

The first angle = 60°

The second angle = 1.2(60) = 72°

The third angle = 0.8(60) = 48°

Hence, the three angles of a triangle are 60°, 72° and 48°.

**Example 6 : **

If 3 consecutive positive integers be the angles of a triangle, then find the three angles of the triangle.

**Solution :**

Let "x" be the first angle.

Then, the second angle = x + 1

The third angle = x + 2

We know that,

the sum of the three angles of a triangle = 180°

x + x + 1 + x + 2 = 180°

3x + 3 = 180°

3x = 177°

x = 59°

The first angle = 59°

The second angle = 59 + 1 = 60°

The third angle = 59 + 2 = 61°

Hence, the three angles of a triangle are 59°, 60° and 61°.

**Example 7 : **

In a triangle, if the second angle is 2 times the first angle and the third angle is 3 times the first angle, find the angles of the triangle.

**Solution :**

Let "x" be the first angle.

Then the second angle = 2x

The third angle = 3x

We know that,

the sum of the three angles of a triangle = 180°

x + 2x + 3x = 180°

6x = 180°

x = 30°

The first angle = 30°

The second angle = 2 ⋅ 30° = 60°

The third angle = 3 ⋅ 30° = 90°

Hence, the three angles of a triangle are 30°, 60° and 90°.

**Example 8 : **

In a right triangle, apart from the right angle, the other two angles are x + 1 and 2x + 5. find the angles of the triangle.

**Solution :**

We know that,

the sum of the three angles of a triangle = 180°

90 + (x + 1) + (2x + 5) = 180°

3x + 6 = 90°

3x = 84°

x = 28°

So, we have

x + 1 = 28 + 1 = 29°

2x + 5 = 2 ⋅ 28 + 5 = 56 + 5 = 61°

Hence, the three angles of a triangle are 90°, 29° and 61°.

**Example 9 : **

In a triangle, if the second angle is 3 times the sum of the first angle and 3 and the third angle is the sum of 2 times the first angle and 3, find the three angles of the triangle.

**Solution :**

Let "x" be the first angle.

Then, the second angle = 3(x + 3)

The third angle = 2x + 3

We know that,

the sum of the three angles of a triangle = 180°

x + 3(x + 3) + 2x + 3 = 180°

x + 3x + 9 + 2x + 3 = 180°

6x + 12 = 180°

6x = 168°

x = 28°

The first angle = 28°

The second angle = 3(28 + 3) = 93°

The third angle = 2 ⋅ 28 + 3 = 59°

Hence, the three angles of a triangle are 28°, 93° and 59°.

**Example 10 : **

In a triangle, the ratio between the first and second angle is 1 : 2 and the third angle is 72. Find the first and second angle of the triangle.

**Solution :**

The ratio of the first angle and second angle is 1 : 2.

Then, the first angle = x

The second angle = 2x

We know that,

the sum of the three angles of a triangle = 180°

x + 2x + 72 = 180°

3x = 108°

x = 36°

The first angle = 36°

The second angle = 2 ⋅ 36° = 72°

Hence, the first angle is 36° and the second angle is 72°.

After having gone through the stuff given above, we hope that the students would have understood the sum of the three angles in a triangle.

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