Question

If three times the supp. of an angle is subtracted from seven times the comp. of the angle, the answer is the same a that obtained by trisecting a right angle. Find the supplement.

Answer 1

Alright, let's start by giving some notation to "an angle." We'll use the Greek letter
`\theta` to denote it.

Next, we'll need to review a few definitions:

- Two supplementary angles add up to a straight angle, or 180°
- Two complementary angles add up to a right angle, or 90°
- Trisecting an angle is splitting a larger angle into three equal smaller angles.

If we call
`\theta`'s supplement
`\beta` and
`\theta`'s complement
`\alpha`, we know that:


`\theta+\beta=180\\ \theta+\alpha=90`

or, if we want to put everything in terms of
`\theta`:


`\beta=180-\theta\\\alpha=90-\theta`

We're given from the problem that
`3\beta` is being subtracted from
`7\alpha`, which in terms of
`\theta` gives us:


`7(90-\theta)-3(180-\theta)`

Next, we're told that this expression is equal to the angle obtained by trisecting a right angle. A right angle is equal to 90°, so trisecting it, we get the angle 90°/3 = 30°.

Putting everything together, we have:


`7(90-\theta)-3(180-\theta)=30`

From there, solve for
`\theta`, but remember that the question asks for the supplement of
`\theta`, not
`\theta` itself. Fortunately, we have an equation from earlier for the supplement,
`\beta`:


`\beta=180-\theta`

Simply put your result from solving for
`\theta` into that equation and solve, and you'll have your answer.