Question

An angle is equal to five times its complement. Determine its measure.

Answer 1

Hint:First of all try to recollect what complementary angles are. Now, assume two angles as x and y and use the given information to make two equations between them and solve it to find the measure of two angles.

Complete step-by-step answer:
Before proceeding with the question, let us first see what complementary angles are.
Complementary Angles: When the sum of two angles is \[{{90}^{o}}\] then the angles are known as complementary angles. In other words, if two angles add up to form a right angle, then these angles are referred to as complementary angles. Hence, we can say that two angles complement each other. If we have angles m and n and both are complement to each other, then we can say that,
\[m+n={{90}^{o}}\]
\[\Rightarrow m={{90}^{o}}-n\]
Now, let us consider our question. First of all, let us assume the given angles as x. Now, we are given that this angle is equal to five times its complement. So, we get,
(Given angle) = 5 (Complement of the given angle)
By substituting the given angle by x, we get,
(x) = 5 (Complement of angle x) …..(i)
We know that if the sum of the two angles is \[{{90}^{o}}\], then they are complement of each other. So, we get,
 (x) + (complement of x) = \[{{90}^{o}}\]
The complement of the angle x = \[\left( {{90}^{o}}-x \right)\]
By substituting the complement of the angle x by \[\left( {{90}^{o}}-x \right)\] in equation (i), we get,
\[x=5\left( {{90}^{o}}-x \right)....\left( ii \right)\]
By simplifying the above equation, we get,
\[x=450-5x\]
By adding 5x on both the sides of the above equation, we get,
\[x+5x=450\]
\[6x=450\]
By dividing both the sides of the above equation by 6, we get,
\[\Rightarrow x=\dfrac{450}{6}\]
\[\Rightarrow x={{75}^{o}}\]
Hence, we get the measure of the angles as \[{{75}^{o}}\].

Note: In this question, students can verify their answer by substituting the value of x in the equation that we have formed initially as follows:
\[x=5\left( {{90}^{o}}-x \right)\]
By substituting \[x={{75}^{o}}\], we get,
\[{{75}^{o}}=5\left( {{90}^{o}}-{{75}^{o}} \right)\]
\[{{75}^{o}}=5\left( {{15}^{o}} \right)\]
\[{{75}^{o}}={{75}^{o}}\]
So, we get, LHS = RHS
Hence, our answer is correct.