SUM OF THE THREE ANGLES OF A TRIANGLE

In any triangle,

Sum of the three angles  =  180°

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°

Because 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°

So, 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°.

So, 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°.

So, 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°.

So, 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°.

So, 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°

So, 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°.

So, 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°.

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

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