Question

(16) 22. Twice the complement of an angle is 50 less than the angle's supplement. Find the measure of the angle.

Answer 1

Let "θ" represent the angle whose measure we need to find.

Complement (c) of θ

If two angles are complementary, it means that their sum is equal to 90 degrees. To determine the measure of the complement of a given angle "θ", you have to calculate the difference between 90 and the said angle. You can express the value of the complement as:

`c=90-\theta`

Supplement (s) of θ

Two angles are supplementary when their sum is equal to 180º. To determine the measure of the complement of a given angle "θ", you have to calculate the difference between 180º and the said angle. You can express the value of the supplement as follows:

`s=180-\theta`

For the angle "θ" we know that "twice the complement", symbolically 2c, is equal to "50 less than the angle's supplement", symbolically s-50.

So that:

`2c=s-50`

Replace the expressions obtained for c and s:

`2(90-\theta)=(180-\theta)-50`

From this expression, we can determine the measure of the angle:

-First, distribute the multiplication on the left side of the expression, and simplify the like terms on the right side:

`\begin{gathered} 2\cdot90-2\cdot\theta=180-\theta-50 \\ 180-2\theta=180-50-\theta \\ 180-2\theta=130-\theta \end{gathered}`

-Second, pass "180" to the right side of the equation by applying the opposite operation "-180" to both sides of it.

Use the same method to pass "-θ" to the left side of the equation:

`\begin{gathered} 180-180-2\theta=130-180-\theta \\ -2\theta=-50-\theta \end{gathered}`
`\begin{gathered} -2\theta+\theta=-50-\theta+\theta \\ -\theta=-50 \end{gathered}`

-Third, multiply both sides of the expression by -1 to reach the measure of θ

`\begin{gathered} (-1)\cdot(-\theta)=(-1)(-50) \\ \theta=50 \end{gathered}`

The measure of the angle is θ=50º