Points, Lines, Angles, and Planes
Length and Measurement
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 ten, 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 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.