the supplement of an angle is 20 degrees less than 3 times its complement. find the ratio of the supplement to the complement.
s = 3 c - 20
s = 180 - a
c = 90 - a
-------------subtract
s - c = 90
so
s = c+90
c+90 = 3 c - 20
2 c = 110
c = 55
then s = 145
s/c = 145/55 = 29/11
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check
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3(55)= 20 = 165 - 20 = 145 yes
a = 90 -c = 35
180 - 35 = 145 yes
To solve this problem, we need to understand the definitions of the terms "supplement," "complement," and "angle."
An angle is a figure formed by two rays sharing a common endpoint called the vertex. The two rays are called the arms of the angle.
The supplement of an angle is the angle that, when added to the original angle, gives a sum of 180 degrees. In other words, the supplement of angle A is 180 - A.
The complement of an angle is the angle that, when added to the original angle, gives a sum of 90 degrees. In other words, the complement of angle A is 90 - A.
Let's solve the problem step by step:
Let the angle be denoted as A.
According to the problem, the supplement of angle A is 20 degrees less than 3 times its complement. Mathematically, we can represent this as:
Supplement of A = 3(times the complement of A) - 20
Using the definitions above, we can rewrite this as:
180 - A = 3(90 - A) - 20
Now, let's solve this equation to find the value of A.
Expanding the equation:
180 - A = 270 - 3A - 20
Combining like terms:
180 - A = 250 - 3A
Moving the "A" terms to one side of the equation:
2A - A = 250 - 180
Simplifying:
A = 70
So the value of angle A is 70 degrees.
Now, let's find the supplement and complement of angle A:
Supplement of A = 180 - A = 180 - 70 = 110 degrees
Complement of A = 90 - A = 90 - 70 = 20 degrees
Finally, to find the ratio of the supplement to the complement, we divide the value of the supplement by the value of the complement:
Ratio = Supplement/Complement = 110/20 = 11/2
Therefore, the ratio of the supplement to the complement is 11/2.