Geometric Proofs: Definition and Format

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  • 0:03 Geometric Proofs
  • 1:06 Parts of a Proof
  • 2:33 How It's Done
  • 4:31 Sample
  • 7:50 Lesson Summary
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Lesson Transcript
Instructor: Jeff Calareso

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

Do you have something to prove? Can you explain why? In this lesson, we'll learn all about geometric proofs, including the parts that comprise a proof.

Geometric Proofs

Here's A. And waaaay over there is Z. How do we get from A to Z? First, there's B, then C, then D, and then, well, you probably know how this goes. But what about in geometry? What if we have to prove two angles are congruent? We need a geometric proof. A geometric proof is a method of determining whether a statement is true or false with the use of logic, facts and deductions.

A proof is kind of like a series of directions from one place to another. There might be different roads you could take, but as long as you obey known laws to get there (avoiding driving across fields and lakes and such), you achieved your goal.

These proofs can be written as a two-column chart or in paragraph form. You see the two-column kind more often. This is a neat, orderly way to walk through the steps. We're more likely to skip a step in the paragraph style, though there's nothing inherently wrong with that format.

Parts of a Proof

A two-column proof has - wait for it - two columns. The column on the left is a list of statements. These are things we know are true. On the right, we have a list of reasons. This is the 'why' for each statement.

Imagine trying to explain something to a 4-year-old. Every statement you make will be followed with 'why?' A proof is kind of like any conversation with a 4-year-old. You have to explain the why behind every statement. Oh, but as much as you want to, you can't say, 'Because I said so.' That doesn't fly with a 4-year-old, and it doesn't fly in a proof.

For example, we could have 'Chocolate is the best ice cream flavor' on the left. That's a statement we know to be true. We explain it on the right with 'Chocolate is the yummiest.' Okay, the only problem with that statement and reason is that I may be on board with them, but they aren't necessarily true.

Each statement and reason pair has its own row. Each proof has as many rows as it takes to prove whatever it is we're proving. Maybe it's 3 rows, maybe it's 30. To use that travel metaphor again, sometimes you're going from your house to the store across town. That's a short proof. Sometimes you're driving across Canada. That's a long proof, eh?

How It's Done

When you need to complete a proof, you'll be given a problem. This isn't like 2x + 7 = 11, solve for x. That's so algebra. This is geometry. You'll possibly be given some statements to help you. Read the problem carefully and see what it tells you.

Maybe it says, 'Runners have low resting heart rates,' and 'Running improves cognitive function.' Then 'Prove that running is the best activity.' Okay, like the chocolate ice cream, this example is totally subjective (though, you know, probably true anyway). But look at that problem. Two of the three sentences aren't the question. They're given statements, like signs on your journey saying 'shortcut here!'

In a proof, what's given can be a big deal. When you start writing your proof, the only reason why that you need for these statements is that they're given. If it said, 'Runners live, on average, 150 years longer than non-runners,' well, that's crazy talk, but it's a given statement, so it's valid in your proof.

Before you jump in, you should draw a picture. Maybe a picture is provided. If not, definitely draw one. Okay, this may not help with the running example, but in geometry, we're talking triangles, squares and whatnot. Think of it like your map. The picture will help you visualize everything. Plus, you can add to the picture to start your reasoning.

Let's say you're trying to prove that two triangles are congruent. You draw the triangles, then start adding marks to indicate what you can figure out. Maybe it's that angles are congruent, or maybe it's sides. Let the picture guide you.

Next, write the statements and reasons. In a proof, take nothing for granted. Remember the 4-year-old. Assume nothing. Include every step of the thought process and the reason why.

Sample Proof

Should we try a quick one? Okay! 'In triangle XYZ, XY is congruent to XZ. Point B lies on YZ and XB bisects YZ. Prove that angle YXB is congruent to angle ZXB.'

Whoa, that's a lot of letters. It would've been awesome if they gave us a picture. So, what do we do? We start by reading the problem carefully, then we draw our picture.

Triangle XYZ

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