Points, Lines, Angles, and Planes
Topics
Introduction to :
When two lines cross, not only do they meet at exactly one point, but they form four corners as well. For example, when you text, "Spotted: N and S looking cozy at 2^{nd} and 74^{th}—do I see a baby bump?" to the Gossip Girl tipline, it would assist the ogling masses if you also point out which of the corners at the intersection they can find the canoodling couple.
In geometry, these corners are called angles and they always occur when two lines, rays, or segments meet at a point. This point is called the vertex of the angle and the lines (or rays or segments) are called the sides of the angle.
Naming angles is quite a bit trickier than naming points and lines. This is because while points and lines are things you can draw, angles are the space between lines. (Another way of thinking about it is that an angle shows how much you have to rotate one line to reach another.) One way to name an angle is simply to slap down a number in this intermediate space and say, "There, I named it."
It's definitely an option, but adding more labels when lines and points are already involved might be messier than a hoarder's apartment. Another, more meaningful (but often less convenient) way of naming an angle is by using three points. For example, ∠ABC means, "the angle formed by AB and BC."
The order of the letters is important. When you read ∠ABC to yourself, you want to think, "Okay, I'm standing at point A. Now I'll walk to point B. Hm, now that I'm here at point B, I feel like turning the corner and going to C." Since you turned the corner at B, that's where the vertex of your angle is. That means the middle point will always be the vertex of the angle, capisce?
Sample Problem
Give five names for the depicted angle.
Since the angle is given a number, we can just call it ∠1. We need to use points for the four other names: ∠QRS, ∠SRQ, ∠QRT, and ∠TRQ. Notice how R is always in the middle because it's the vertex.
There are a few special kinds of angles worth mentioning. If two angles are next to each other, they're called adjacent angles. More precisely, they must share a vertex and a side, and not share any interior points. We like to think of two slices of pepperoni pizza next to each other in the box (because it's both helpful and tasty).
Here comes the Dan Brown plot twist where angles turn into demons. No, not angels. Angles.
It's possible for angles to share a side but not be adjacent. Say what? This often happens when cartoon lightning strikes.
That means ∠BAN and ∠ANG are not adjacent because they have different vertices. You're more likely to be struck by cartoon lightning 7 times than win the cartoon lottery, so that's useful information.
Also, if one angle is inside the other, they don't count as adjacent.
For example, ∠ABC and ∠ABD are not adjacent since ∠ABC is sitting inside ∠ABD. (To be fair, it did call a seat check). However, ∠ABC and ∠CBD are adjacent. Make sense?
Careful, because there are more demons headed our way. Maybe we should get some bug repellant or something.
In possibly the worst named geometrical term of all time, two angles are vertical if they're opposite each other in a configuration of two intersecting lines. As with adjacent angles, vertical angles always share a vertex, but they rarely share a side.
Why are they called vertical angles if they can be diagonal or even horizontal? We agree. They should be called vertical demons instead.
Sample Problem
Which angles in this figure are vertical angles?
Vertical angles are really just angles opposite one another. Looking at the intersecting segments, we can see that ∠NCW and ∠ECS are vertical angles, and so are ∠NCE and ∠WCS.
Sample Problem
Which angles in the same figure are adjacent angles?
Adjacent just means they share a vertex and a side. If we count them, there are four pairs of adjacent angles: ∠ECS and ∠SCW, ∠SCW and ∠WCN, ∠WCN and ∠NCE, and ∠NCE and ∠ECS. If sharing is caring, they must really care for each other.
If we wanted to be super nitpicky, we could also say that technically, angles like ∠ECW and ∠ECS are also adjacent. That would add 4 adjacent angles for ∠ECW and another 4 for ∠NCS to make a total of 12 adjacent angles. We can't say that ∠ECW and ∠WCE are adjacent, though. Why? Because they're the same angle.
Example 1
What is the vertex of ∠1?

Example 2
Give four names for ∠1 (aside from ∠1, obviously).

Example 3
Name an angle adjacent to ∠1.

Example 4
Name a pair of vertical angles.
