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Angles: Types and Labeling - Concept

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

There are four types of angles: acute, right, obtuse, and straight. Each name indicates a specific range of degree measurements. Congruent angles have equivalent measures. Adjacent angles share a vertex and a common side.

An Angle is something that we use throughout Geometry, we talk about it all the way to the very end when we're talking about Trigonometry. Well an angle is formed by two rays that share common end point. It's measured in degrees and is between 0 and 180 degrees, if it's over 180 then you're going to subtract that number. So let's say you had 220 degrees you're going to subtract 180 from that so it's actually a 40 degree angle.
If we look at an example where we have angle a, b, c, there's two ways that you could label this. You can write this as angle abc or since there are no other adjacent angles that is an adjacent angle would be something like this where it would share that vertex that common end point. Since there are no other adjacent angles you could also just label this based on the vertex which is b.
Now there's something very specific about the way that I wrote angle abc, whenever you write the angle its vertex must be the middle letter. But what is the vertex? The vertex is this point that is the common end point of your rays. So I'm going to label this as the vertex, so the rays form what are called the sides. So bc, ray bc is one side of this angle and ray ba is another side. So again you can label an angle two different ways, one using three letters that make up the two sides and the vertex making sure that your vertex is the middle letter or if there aren't any other adjacent angles you can just label it based on its vertex.
There are four key types of angles first one is a acute, so if I drew this angle and I said that's angle x if it is acute that means that is less than 90 degrees but also greater than 0 degrees. So it has to be somewhere in between them, it cannot be exactly 90 degrees it cannot be exactly 0 degrees.
A right angle if this is x is equal to exactly 90 degrees a right angle. So we're going going to label all of our right angles in Geometry using these two segments which will tell you the student that this is a 90 degree angle.
The third type is an obtuse angle. So here if we measured x, x is going to be less than 180 degrees but more than 90 degrees. Because if this angle was able to be exactly 90 degrees it would be a right angle if it was less than 90 it would be acute.
The last one which is key to a lot of proofs is a straight angle. If you have a straight angle it is the equal to exactly 180 degrees. Which means if we had the rotation about point x that was form the full rotation around any given point is 360 degrees. So 2x=360 degrees.
So keep this in mind and remember obtuse is going to be in between 180 and 90, right is going to be exactly 90, acute is going to be between 90 and 0. And that there are two different ways of labeling your angles. And the way that will always work is using three letters.

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