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Before we can talk about objects or concepts in geometry, we need actual material to work with. Without them, we're no better than a lumberjack without any lumber. Would that just make him a jack?
Well, it's easy for lumberjacks. They can go out and get what they need without a second thought. When discussing geometry, on the other hand, we might not even have a clue where to start. It's like a big empty void of nothingness, and we're expected to somehow create these building blocks of geometry using only logic.
Fine, then. We'll make these basic units of geometry—the point, the line, and the plane—and call them undefined notions. Since there was nothing that existed before them, we can't use anything to define them—but that doesn't mean we can't explain them.
What's the smallest thing you can think of? A grain of sand? An atom? Rhode Island?
Actually, a point is even smaller. Even though it's one of the most fundamental objects in geometry, the point is just barely there. It's the tiniest object imaginable. It has no size, no mass, no nothing. All it has is a location.
For example, the center of this circle is a point. We typically draw points with little dots, but a point itself is even smaller than a dot (and smaller than Rhode Island…probably).
Even though some points are known for being breathtaking national parks or just being pleasant, mathematicians usually name their points a single, capital letter. How creative. So in the above picture, we'd simply call the point P. Hi, P.
Imagine now that you're standing on point A, and your friend on point B, and you're both tugging on a rope (don't ask us what's so exciting about this rope—you're the one tugging on it!). The rope will end up going straight between the two of you every time you repeat this process.
You and your friend have ingeniously shown that connecting two points gives a line segment. That's because there's only one way to join two points without curves or corners. Give yourselves a round of applause.
If you can't stop yourself and extend the segment in both directions forever, off the page, off the planet, and to the ends of the universe, we'll get a line. Actually drawing a line would use up a lot of ink, so we'll use little arrows to show that it goes on forever instead.
Finally, if you extend the segment in only one direction, you end up with a ray, which looks like this:
Think "ray of sunshine," not "stingray."
The thing to remember in all this is that any two points make exactly one line segment and exactly one line. They give two rays, depending on which endpoint you extend and which you don't. We don't really have a preference, so we'll leave that up to you.
That means we can write lines using names for the points involved, like so.
Sometimes we instead use single, lowercase letters to name lines, so we can call this guy XY or YX or just l. Don't call him Twiggy, though. He hates that.
Even though lines contain several points (in fact, they contain infinitely many), we get tired after drawing about ten or so. Instead, we'll use 2 points to name a line. That's it.
So this line can go by RS or ST or TS, but NOT RST. Putting in extra points is redundant, and if there's one thing that mathematicians could get rid of in this world, it's redundancy. Either that or the absurdly long lines at Disneyland.
Here's a video on graphing points in a grid:
How many ways are there to draw a line segment on a piece of paper with both endpoints at corners of the page?
First, we'll make each of the corners a point, giving them names like A, B, C, and D. We know that for any two corners, there is exactly one segment connecting them, and those are the only segments we care about.
Writing the possible ways of choosing two corners, we get AB, AC, AD, BC, BD, and CD. These six segments are the four edges of the page and the two diagonals. Geometry is all about pictures, so here ya go.
What about lines? How many ways are there to draw lines using the four corners?
The answer is still six, but the lines have arrows at their ends because they extend to infinity and beyond. We can still write them as AB, AC, AD, BC, BD, and CD, though.
Some of the lines cross (BD and AC cross in the middle of the page) and some of the lines never cross (AD and BC, for instance). If lines on the same page never cross, we call them parallel, but we'll talk about them later. For now, we should give intersecting lines more attention.
When two lines cross, we say they intersect, which means they overlap at exactly one point. This makes sense if we draw it, since to get one line to cross another twice we end up having to curve the line or make a corner.
Lines are supposed to be straight, so that's a no-go…or is it? After all, just because we can draw it doesn't make it logically true. We could draw a picture of a dragon eating a unicorn listening to Lindsay Lohan's Speak, but that doesn't magically mean any living being actually bought that album.
This (the intersecting lines, not the unicorn-eating, Lohan-loving dragon) brings us to our very first proof in geometry: deducing logically that intersecting lines cross at exactly one point. We'll use the "proof by contradiction" technique. Sounds snazzy, but it just means we're going to prove it by saying that the opposite can't possibly happen. Some call this an indirect proof because we're arriving at our conclusion indirectly.
Suppose that two different lines l and m cross at points A and B. Since any two points determine a line, we know l is the same as AB and m is the same as AB. But then that means l is the same as m, which contradicts our assumption that they were different lines.
Some prefer the "two column proof" format instead of the informal argument we used. We're going to be stuck introducing concepts for a while, so there aren't going to be many formal proofs right away. Don't be sad, though. We have a lot of more complicated proofs to look forward to later on.
All these points and lines remind us of connect-the-dots. Remember playing that as a kid? Well, if we imagine the most boring game of connect-the-dots ever, it would probably look something like this.
All we're really doing is extending the segment along the same straight line. This is what it means for points to be collinear: we can draw a single line through all of them at once. Of course, forming a line segment isn't exactly what we're signing up for when we play connect-the-dots. That's why the dots usually aren't collinear.
For example, there's no way of drawing a single line through points X, Y, and Z. (Don't try to draw a fat line, either, because lines have no width or depth, only length.)
On the other hand, we can clearly see that the segment AC passes directly through B, so the points are collinear. Instead of saying "A, B, and C are collinear," we could say, "B is between A and C." They mean the same thing, but the second one tells us that B is in the middle.
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 2nd and 74th—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?
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.
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.
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.
We can't say that ∠ECW and ∠WCE are adjacent, though. Why? Because they're the same angle.
What happens when segments and angles stop being polite, and start getting real? Step back, Jersey Shore. There's a new reality show in town. As fascinating as a geometry-themed TV series sounds, we're talking more about using geometrical objects in the real world (no, not that "The Real World").
One of the most ancient applications of geometry is in architecture, and one of the most important parts of turning blueprints into a building is the accurate measurement of the building blocks. Without measurement, the Parthenon might have been more of a Parthenot.
What we're trying to say is that segments have length, which is just the distance between their endpoints. Lengths can be measured in several units: inches, feet, meters, furlongs, parsecs, light-years, whatever.
When building space-bound machinery (like the Wolowitz Zero-Gravity Human Waste Disposal System), it's important to get the exact units right. We aren't NASA engineers, so the exact units aren't critical to us. The point is that all segments with the same length look pretty much the same. See?
So, if you're building a temple to some groaning Greek goddess, and the plans call for a thirty-foot marble column, it doesn't matter which of the dozen columns your slaves are toting behind you that you pick. Any thirty-foot column will do as long as you put it in the right place.
Similarly, we have a special name for segments that are the same length. We call them congruent (and we use the symbol ≅). Just like columns, if one segment is congruent to another and we move them to the same place, they look exactly the same (and are mathematically equal).
Angles are a bit trickier to measure. There are a couple different units people use, but degrees are probably the most common. The idea behind the degree is to chop up a circle into a bunch of tiny wedges, and see how many of these wedges fit into the angle you want to measure. (Of course, nowadays it's easier to just use a protractor.)
There are 360 degrees in a circle, declared so by Tony Hawk the first time he spun around on a skateboard.
Actually, the Babylonians had a complicated system based on 60 rather than 10, so we have them to thank for the weird number of degrees. They're also the reason we have 60 seconds in a minute. Way to go, Babylonians.
Right angles are the most special of all. Don't tell the other angles though. We don't want them to feel bad. A right angle measures exactly 90 degrees and looks like a corner of a piece of paper.
In symbols, we'd say m∠3 = 90° or m∠ABC = 90°. When we want to drive home the point that an angle is a right angle, we can put a little box in the corner. That way, if we see a box in an angle, we'll automatically know that it's 90 degrees.
Two lines or segments that form a 90-degree angle are said to be perpendicular. Yeah, it's quite a mouthful. Perpendicular lines are related to parallel lines, so we'll also talk about them some more later on.
Just like with segments, we call two angles congruent if the have the same measure. If you're bored, you can draw a bunch of angles of the same measure on a piece of paper and check that you can put one on top of another without any funny business. It's not our idea of a good time, really, but who are we to judge?
Why do we care about angles that are all the same, anyway? Well, because sometimes, congruent angles might not look the same. Take vertical angles like ∠LMN and ∠OMP, for example.
Whether it's obvious or not, it's important to know that vertical angles are always congruent. Always, always, all-of-the-ways, always.
This is probably the most important thing we've said so far. Many of the problems we'll do involving angles rely on this crucial fact, and here's an example of how to use it.
In the figure above, m∠OMP = 46°. What is m∠LMN?
As we mentioned, ∠OMP and ∠LMN are vertical angles (formed by the intersection of lines LP and NO). Since vertical angles are always congruent, that means m∠LMN = 46°, too.
The notions of angles and lines also intersect. (Note the brilliant use of the word "intersect.") Yes, that's right. Straight lines make angles.
Imagine you smell a rose and hear a mortifying buzz behind you. In your last moments before being swarmed by killer bees, you turn around to get a glimpse of the droning horde.
The bees, you, and the rose are collinear. We think of this as an angle with you at the vertex, so when you turn from the roses to the bees, you make a 180° angle. If you keep turning back to the rose (in order to run away as fast as humanly possible), you'd make a 360° angle, a full circle. See, we told you geometry has real-world (and potentially life-saving) applications.
Angles that are smaller than 90° are called acute 'cause they're just so darn adorable. Angles greater than 90° are called obtuse even though they're just as intelligent as other angles. Angles that are 180° in measure are said to be straight angles even though they're really just lines.