Math CentralQuandaries & Queries


Question from tom, a student:

The measure of the complement is 12 more than twice the original angle. Find the measure of the angles.

Hi Tom.

The measure of the complement of any angle A is ninety degrees minus the measure of angle A.

So if you have an angle that is 20 degrees, then the complement angle measures 70 degrees.

Now let's parse your question, starting by defining our variables:

Let C = the measure of the complement angle and let A = the measure of the original angle.

"The measure of the complement is..."
C =

"...12 more than..."
C = 12 +

"twice the original angle."
C = 12 + 2A.

Okay, but that has two unknown values. However, we know that A + C = 90 as well.

So we have two equations with two unknowns. If you have the same number of unique equations and unknowns, you can solve the problem.

Remember that both equations are true at the same time, so C = 12 + 2A at the same time A + C = 90. So if C = 12 + 2A, then I can replace one mathematical quantity (C) with what it equals (12 + 2A) anywhere I like. I will replace the C in the second equation.


A + C = 90
turns into
A + (12 + 2A) = 90.

Now you can solve for A and you will know what the original angle measurement is. Use that to determine the complement's measurement.

Stephen La Rocque.

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