Question

Two complementary angles differ by 12${}^{\circ }$, then find the angles.

Answer 1

If two angles are complementary, the sum of their measures is $90$ degrees.

Let one angle be $x$.

This means that its complement is $90-x$ degrees.

But the difference in their measures is $12$ degrees.

Therefore, you can write the equation:

$x-(90-x)=12$

Since the parentheses are preceded by a minus sign, you can remove the parentheses if you also change the signs of the terms inside the parentheses. Therefore, when you remove the parentheses

the equation becomes:

$x-90+x=12$

Combine the two $x$ terms on the left side to get:

$2x-90=12$

Get rid of the $-90$ on the left side by adding $90$ to both sides to get

$2x=102$

Solve for x by dividing both sides of the equation by $2$ to get:

$x=\frac{102}{2}=51$

So, the measure of one of the two angles is $51$ degrees. Its complement is the amount that

must be added to that angle such that this sum is $90$ degrees. By subtracting $51$ from $90$

degrees, you find that the complement of the angle is an angle of $39$ degrees.

Note that

the difference in the two angles is $51-39=12$ degrees, just as the problem specified.

The answer to this problem is that the measures of the two complementary angles are $51$

degrees and $39$ degrees.

Hence, this is the answer.