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Logic and Proof

Logic and Proof

Geometric Proofs

One of the biggest differences between proofs in algebra and proofs in geometry is that geometrical proofs have pictures. By pictures, we mean images of geometric shapes, not lolcats.

Since geometry is concerned with things you can draw, like points, lines, angles, and the like, translating pictures into proofs and vice-versa can't really be avoided. That's okay, though. They're more interesting to look at than endless lines of text. Even this picture of a triangle gets your adrenaline pumping.

ABC is made of three segments: AB, BC, and AC. Also, we can see that they make three angles: ∠BAC, ∠ABC, and ∠ACB. If that's already too much geometry lingo for you, get a quick refresher or a more in-depth look.

We can add little tick marks (be careful of Lyme disease!) to show that segments are congruent, which means they're exactly equal in measurement. For example, if we mark both AB and BC with a single tick mark like so:

It means that AB and BC are the same length, or ABBC. They're congruent (expressed by that equal sign with the squiggly on top). We can do the same thing with angles.

Now ∠ABC and ∠ACB are marked congruent as well. By adding even more tick marks, we can account for several different congruencies.

Sample Problem

Draw a figure with points P, Q, R, S and segments such that PQRS and QRPS.

First we draw the points and the segments:

Then we mark the first congruence with a single tick mark:

Finally, we mark the second congruence, this time with double tick marks (so that we don't confuse all four segments as being congruent).

In some instances, there are several ways to draw the same diagram. For example, if we placed the points like this:

and then drew the segments, we'd end up with some crossing segments.

There's nothing wrong with this picture, but it is a little harder to work with. For example, the next step is to put tick marks to show the congruence of PQ and RS.

It's hard to tell whether we're saying that the big segments are congruent or just the little parts before the intersection. In general, it's a good idea to try to draw diagrams so that no lines cross, if possible.

If they do cross, have no fear. Two crossing segments make vertical angles, like the ones here.

Vertical angles are congruent, but it's not always necessary to draw their tick marks. It should be obvious from the image that they're opposite angles from an intersection of two segments.

When only part of a segment sticks out from the middle of a different one (like a leaning palm tree sticking out of the sand), the two angles are called supplementary. That means they both add up to 180°. We'll learn more about vertical and supplementary angles and all that good stuff later on. For now, just take our word for it.

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