If two angles are complementary and the sum of the measure of the first angle and one-fourth the measure of the second angle is 78 degrees what are measures of the angles?

Answer 1

12

Answer 2

Setting up the equationsYou need to read the text of the question carefully and then set up equations that describe the words mathematically.

So, let's break it down piece by piece.

"Two angles"

Let's label the angles A and B. (You could have called them X and Y. It makes no difference what you call them. You could have called them Moe and Curly.)

"are complementary"

Two angles are complementary when their sum equals 90 degrees. So, if A and B are complementary, then A + B = 90. (That looks like an equation, doesn't it?)

"one fourth the measure of the second angle"

If we let A be the first angle, then that means B is the second angle. One fourth (the measure) of the second angle can be written 0.25B or (1/4)B or B/4. For this problem, I like 0.25B

"the sum of the (measure of the) first angle and one fourth the (measure of the) second equals 78 degrees"

"Sum" implies addition. So we have A + 0.25B = 78. (That looks like another equation doesn't it?)

Since we now have two equations, we can solve for the two unknowns, A and B.

The first equation is A + B = 90. If you solve for A in terms of B, you get A = 90 - B. (If that has you puzzled, you'll need to brush up on your basic algebra.)

Now, substitute 90 - B wherever you see an "A" in the second equation, so A + 0.25B = 78 becomes 90 - B + 0.25B = 78.

Next, group like terms to get 90 - 0.75B = 78.

Now solve for B. First, subtract 78 from both sides of the equation to get 12 - 0.75B = 0. Next, add 0.75B to both sides to get 12 = 0.75B. Finally, divide both sides by 0.75, which gives you 16 = B, or B = 16.

If B = 16, then A = 90 - B = 90 - 16 = 74 degrees. Your two angles, A and B, are 74 and 16, respectively. Does that make sense? Let's see. Both angles add up to 90, which means they are complementary. And A plus one fourth of B is 74 plus 4 equals 78. Yep. We re now very good to go.

Thank you everybody and have a nice day. :)