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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