**Sum of the three angles of a triangle:**

Irrespective of the type of the triangle,

Sum of the three angles in any triangle = 180°

More clearly,

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

**Problem 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°

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

Let us look at the next problem on "Sum of the three angles of a triangle"

**Problem 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. **

**Let us look at the next problem on "Sum of the three angles of a triangle" **

**Problem 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.

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°. **

**Let us look at the next problem on "Sum of the three angles of a triangle" **

**Problem 4 : **

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

**Solution :**

From the ratio 2 : 7 : 11,

the three angles are 2x, 7x, 11x

In any triangle, sum of the angles = 180°

So, 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°, 99°)**

**Problem 5 : **

In a triangle, If the second angle is 10% 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.

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°. **

**Problem 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.

The second angle = x + 1

The third angle = x + 1 + 1 = 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 = 60 + 1 = 61°

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

**Problem 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.

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°. **

**Problem 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, 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°. **

**Problem 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.

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°. **

**Problem 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 :**

From the given ratio,

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°. **

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