Question

The measure of an angle is \[28{}^\circ \] greater than its complement. How do you find the measure of each angle?

Answer 1

Hint: This question belongs to the topic of algebra. For solving this type of question, we should know that the sum of an angle and the complement of the same angle is always equal to\[90{}^\circ \]. In this question, first we write the equation of adding the angle and its complement which will be equal to 90 degrees. After that we will write the equation in mathematical form from the phrase “the measure of angle is 28 degrees greater than its component”. After that, we will solve both the equations.

Complete step-by-step answer:
Let us solve this question.
 In this question, it is said that the measure of angle which we have to find is \[28{}^\circ \] greater than its complement. We have to find the measure of angle and its complement.
Let the angle be x and the complement of the same angle be y.
So, we can say from the question that if we have two angles (suppose x and y) and they are complement of each other, then their sum will be of 90 degree (or\[90{}^\circ \]). And, one of x and y is greater than the other by\[28{}^\circ \].
Therefore, for the complementary of the angle, we can write from the above lines
\[x+y=90{}^\circ ...............(1)\]
And, also it is given that the angle x which is 28 degrees (\[28{}^\circ \]) greater than the complement of the same angle x. Here, we have taken y as the complementary angle.
So, we can write
\[x=y+28{}^\circ .................(2)\]
Now, we will solve the equation (1) and (2) to get the value of x and y, so that we will get the value of the angle of its component.
By adding equation (1) and (2), we get
\[x+y+x=90{}^\circ +y+28{}^\circ \]
The above equation can also be written as
\[\Rightarrow x+y+x-y=90{}^\circ +28{}^\circ \]
We can write the above equation as
\[\Rightarrow x+x=90{}^\circ +28{}^\circ \]
The above equation can also be written as
\[\Rightarrow 2x=118{}^\circ \]
The above equation can also be written as
\[\Rightarrow x=\dfrac{118{}^\circ }{2}=59{}^\circ \]
Hence, the value of x is\[59{}^\circ \].
Now, we can find the value of y by putting the value of x in equation (2).
So, we can write from the equation (2) after putting the value of x as\[59{}^\circ \], we will get
\[59{}^\circ =y+28{}^\circ \]
The above equation can be written as
\[\Rightarrow 59{}^\circ -28{}^\circ =y\]
Hence, from the above equation, we can write the value of y as
\[\Rightarrow y=31{}^\circ \]
Therefore, the value of angle is \[59{}^\circ \] and its complement value is \[31{}^\circ \].

Note: We can solve this question by different methods.
As, we have found the two equations in the above, those are
\[x+y=90{}^\circ ...............(1)\]
And
\[x=y+28{}^\circ .................(2)\]
Here, we can solve this equation by a shortcut method. We will take the value of x from equation (2) and put the value of x in equation (1), we get
\[y+28{}^\circ +y=90{}^\circ \]
The above equation can also be written as
\[\Rightarrow y+y=90{}^\circ -28{}^\circ \]
We can write the above equation as
\[\Rightarrow 2y=62{}^\circ \]
From here, we get that
\[\Rightarrow y=\dfrac{62{}^\circ }{2}=31{}^\circ \]
So, the complement of angle is \[31{}^\circ \] and the angle will be \[90{}^\circ -31{}^\circ =59{}^\circ \]