Angles & Their Measures

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.

stands for an angle. You can put it in front of three 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. angle ABC

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's only if there are no other angles that share the same vertex. There is a third way to label angles. In the third way, each angle is designated with a number, so the example could be labeled

1 or

2 or whatever you wanted.

Angles are measured in degrees. The number of degrees tell you how wide open the angle is. You can measure angles with a protracter, and you can buy them at just about any store that carries school items. Degrees are marked by a ° symbol. For those of you whose browsers can't interpret that, a degree symbol looks like this: . I tend to just write it out instead of using the symbol because it's quicker on the computer. There are up to 360 degrees in an angle. As you can see in the picture below, 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. 360 degrees The next important type of angle is called the acute angle. An acute angle is an angle whose measure is inbetween 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 spifty things that you can do with one. I won't go into all of them here. (I have to save something for later articles!) A right angle is an angle whose measure is exactly 90 degrees. Continuing around the circle, next is the obtuse angle. An obtuse angle is an angle whose measure is inbetween 90 and 180 degrees. The 135 degree angle in the diagram is an example. The last major kind of angle is the straight angle. A straight angle is an angle that measures exactly 180 degrees. Thus the name - the two rays form a straight line. A negative angle is also possible. This just means that you go clockwise instead of counterclockwise.

A lot of geometry teachers don't go beyond that, at least at first. There isn't much else left to explain, but I'll give it a shot. After straight angles, there aren't any more special angles that you need to know about. A 360 degree angle is an angle that does a full circle. It looks just like a zero angle, but instead of having no degrees, it has 360 of them. (Duh. You can't get more basic than that!)

It is possible to have an angle with more than 360 degrees. To find out what it looks like, all you do is subtract 360 from it until you have an angle less than or equal to 360. (What?! You want an example? C'mon, you people...) For example, if you have an angle that is 546 degrees, you subtract 360 from 546 to get 186. Thus, the angle is the equivalent of a 186 degree angle.

There are a few more terms that you should also know. Supplementary angles are two angles whose measures combined equal 180 degrees. Complementary angles are two angles whose measures combined equal 90 degrees. Two non-straight and non-zero angles are adjacent if and only if a common side is in the interior of the angle formed by the non-common sides. A linear pair is a pair of adjacent angles whose non-common sides are opposite rays. Vertical angles are two angles that have a common vertex and whose sides form two lines. ray AB is a bisector of DAC if and only if ray AB is in the interior of DAC and mDAB = mCAB.

Angle Measure Postulate

A) Unique Measure Assumption: Every angle has a unique measure. The measure could be infinite or negative.
B) Two Sides of a Line Assumption: Given any ray and any number x, there are unique rays and such that intersects ray and mBEA = mCEA = x.
C) Zero Angle Assumption: If and are the same ray, then mAEB = 0.
D) Straight Angle Assumption: If and are opposite rays, then mAEB = 180.
E) Angle Addition Assumption: If (except for point E) is in the interior of AEB, then mAEC + mCEB = mAEB.

Theorums

Linear Pair Theorum: If two angles form a linear pair, then they are supplementary. See proof.
Vertical Angle Theorum: If two angles are vertical angles, then they have equal measures. See proof.

Note: While you can usually get away with not knowing the names of theorums, your Geometry teacher will generally require you to know them.

- Jaime III
We're really not this boring in person. Honest!