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Lesson 124: Supplementary, Complementary, and Vertical Angles
In this lesson we'll do some geometry work with basic angles, and we'll learn some new definitions. These terms come up frequently, so make sure you understand them.
Before starting this lesson, make sure that you review all of the previous lessons that we've had about angles, and make sure that you fully understand them.
This lesson has a lot of definitions and concepts that need to be memorized. Very often a test question is designed to simply see if you understand what a term means.
First, remember from previous lessons that a right angle is defined as having 90°, and a straight angle (a straight line) is defined as having 180°. It's important to understand where these numbers come from. The degree measure of an angle is really the amount of rotation that is necessary to get from one ray to the other. If the rotation goes all the way around to make a full circle, we define that as 360°. From that, we can see that a right angle is actually one-quarter of a circle, or 90°. A straight angle is half a circle rotation, or 180°. A 36° angle would be 36/360 or one-tenth of a circle rotation.
Take
a look at the two angles at right. You can see that combined,
they make a right angle. That means that whatever each angle
measures, we know that the sum of the two angles is 90°.
We define such a pair of angles to be complementary.
That term needs to be memorized. It could be
said that one such angle is the complement of the other.
Sometimes a word problem can be based around this definition. Here is an example. Two angles are complementary. One measures 3x + 10. The other measures 7x - 20. What is the value of x? We can solve this by setting up an equation as follows: 3x + 10 + 7x - 20 = 90. We know that the sum of the two angles must equal 90. Using basic algebra like we've learned, we can see that the value of x is 10. If we were asked to determine the actual angle measurements, we could substitute our value of x, and see that the angles measure 40° and 50°.
Let's
take a look at the parallel lines and transversal that we've seen
before. There are many different types of angles that
are formed. First, you can see that angles A and B lie along a
straight angle. That means that whatever
each of those angles measures, we know that their sum is 180°.
We call such angles supplementary. That term
needs to be memorized. It could be said that
one such angle is the the supplement of the other.
As above, we can easily solve algebra problems involving supplementary angles, as long we we remember that they their sum is 180°.
Look at angles F and G. These angles are called opposite, or sometimes vertical. These terms need to be memorized. Notice that vertical angles are equal in measure. E and H are also vertical angles, and you can see that they are equal in measure. Knowing this, we can solve associated algebra problems. For example: Two angles are vertical. They measure 2x + 20 and 4x - 10. What is the measure of each angle? All we need to do is set up an equation, and set these two expressions equal to each other, as such: 2x + 20 = 4x - 10. Using basic algebra like we've learned, we can find that x = 15. Substitute that value into either expression, we can see that each angle equals 50°.
You'll learn more about angles in the next lesson.
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