Angles & Their Measures

Understanding Angles

A definition of an angle would be that an angle is the union of two rays that have the same endpoint. The sides of the angles are the two rays, while the vertex is their common endpoint. We use letters which represent points. The first and third letters represent points on each of the rays that form one of the sides. The middle letter represents the vertex.

As you can see in the diagram, each point is represented in the written form. The letters can go either way - that is, first and last letter are interchangeable. So, that angle could either be ABC or CBA. Since B is the vertex, it is always in the middle of the two letters. You can also name an angle by just the letter of its vertex. So, for the example in the picture, the angle could also be labeled B. That is only if there are no other angles that share the same vertex.

Angles are measured in degrees. The number of degrees tell you how wide open the angle is (you can measure angles with a protractor). Degrees are marked by a ° symbol. There are up to 360 degrees in an angle. As you can see in the picture, the 360 degrees form a circle. There are a few more basic things you should know about angles. First of all, the space inside an angle of less than 180 degrees is a convex set, while the space outside of one is a nonconvex set. The opposite is true for an angle of more than 180 degrees (but less than 360 degrees).

The side of an angle that is started at would be called the initial side, and the side that an angle ended at would be called the terminal side. The measure of ABC is written mABC.

When measuring angles, you usually go counterclockwise, starting where the 3 would be on a clock. That would be called a zero angle because there is nothing in it - just a single ray going directly to the right. The next important type of angle is called the acute angle. An acute angle is an angle whose measure is in between 0 and 90 degrees. An example would be the 45 degree angle in the picture. The next important type of angle is the right angle. This is probably the most important type of angle there is because of all the things that you can do with one. A right angle is an angle whose measure is exactly 90 degrees.
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Continuing around the circle, next is the obtuse angle. An obtuse angle is an angle whose measure is between 90 and 180 degrees (the 135 degree angle in the diagram for example). The last major kind of angle is the straight angle. A straight angle is an angle that measures exactly 180 degrees (the two rays form a straight line). A negative angle is also possible when you go clockwise instead of counterclockwise.
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Here are a few more terms that you should know. Supplementary angles are two angles whose measures combined equal 180 degrees. Complementary angles are two angles whose measures combined equal 90 degrees.
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For more information about angles and other mathematical concepts, check out these sites on the WorldWideWeb
InteractiveLearningNetwork / Mathforum / AcademicInfoNet