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The other day I was doing an online session with one of my students on complementary and supplementary angles. He wanted me to help him solve some problems that I knew could easily be solved using algebraic equations. However, the problem was, he did not learn how to frame and solve equations in algebra. So I made him use one of my favorite techniques (more on that a little later).
For the starters, let me explain what complementary and supplementary angles are.
Two angles are complementary if they add up to 90°. For example, 30° and 60° are complementary (or complements of each other) because 30° + 60° = 90°. Similarly, 48° and 42° are complementary angles.
50° and 60° are not complementary angles because 50° + 60° ≠ 90°.
Two angles are supplementary if they add up to 180°. For example, 50° and 130° are supplementary (or supplements of each other) because 50° + 130° = 180°. Similarly, 75° and 105° are supplementary angles.
45° and 65° are not supplementary angles because 45° + 65° ≠ 180°.
Let's say there are two numbers that add up to 100 and one of the numbers is 10 more than the other one. Can you find the numbers?
If you know how to set up and solve simple equations in Algebra, this is as easy as drinking a glass of water. You would assign a letter, say x, to one of the numbers, say the smaller one so that the larger one is x + 10 (that is, 10 more than x).
So you have the equation x + (x +10) = 100. Solving for x gives x = 45. So the numbers are 45 and 55.
The question is, how do you explain this to someone who doesn't know how to set the equation up (and then solve it)?
The trick is to start by splitting the sum in two equal halves, 50 and 50 in this case, as the sum is 100. If the two numbers are 50 and 50, the difference between them is 0. We need to create a difference of 10 between them. The idea is, increase one 50 by an amount and decrease the other 50 by the same amount, so that the sum remains fixed at 100. And because you need to create a difference of 10, you want to increase one by 5 (that is, half of 10) and decrease the other by 5. So the two numbers are 50 - 5 = 45 and 50 + 5 = 55.
This method not only helps you solve problems easily, but you can actually do them mentally. Watch this idea put to use in the following lesson.
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a) Two angles are complementary. If the larger angle measures 12° more than the smaller one, find the measure of the larger angle.
(Ans: 51°)
b) Two angles are supplementary. If the larger angle measures four times as much as the smaller angle, find the measure of the smaller angle.
(Ans: 36°)
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