SUM OF THE THREE ANGLES OF A TRIANGLE

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.

Examples

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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