About this video
Sal explains three building blocks of geometry: postulates (ideas we assume are true), conjectures (ideas we suspect are true from experience) and theorems (conjectures that have been proved). He then uses the linear pair postulate and the transitive property to prove the conjecture that vertical angles are congruent, turning it into a theorem.
You'll be able to
- explain the difference between a postulate, a conjecture and a theorem.
- use the linear pair postulate to find a missing angle.
- explain why repeated measurements alone do not prove a conjecture.
- follow a proof that vertical angles have equal measures.
Key words
- postulate
- A basic statement we assume is true without proof so we can build other ideas on it.
- conjecture
- An idea that seems to be true, often from noticing a pattern, but has not been proved yet.
- theorem
- A statement that has been proved to be true using postulates and other proved facts.
- transitive property of equality
- If a = b and b = c, then a = c.
- linear pair
- Two angles side by side whose outer sides form a straight line, so they add up to 180 degrees.
- vertical angles
- The pair of angles exactly opposite each other where two lines cross.
Transcript
- [Instructor] In this video, we're gonna think about some of the most basic and important ideas in geometry, and that's postulates, conjectures and theorems. And they sound very fancy. But by the end of this video, they will hopefully make a lot of sense. So let's just start with postulates.
Postulates are things that we just assume to be true. Because as we'll see when you eventually have to go about proving something, you have to assume some basic things are true. And then from those, you can build on it. You can logically build on it to prove other things are true.
So what's an example of a postulate? Well, one that you might have already seen in mathematics is the transitive property. Sometimes it's called the transitive property of equality. And that's that if a equals b and b is equal to c, then a is equal to c. That might be common sense to you.
But we like to write things down in mathematics to make sure we are showing people the things that we are assuming. So this right over here, the transitive property, that's a postulate that we are just going to assume. I'll give an example of one that's definitely in geometry.
So it could be the linear pair postulate. And that sounds very fancy. But once again, this must... This might be something that you are already familiar with. That's the idea that if I have two angles right over here, let's call that angle a and angle b. Or let me say that this is a degrees and that this is b degrees, that the sum of these angles.
These angles are in a linear pair because notice that these outsides parts of it form a line, that these add up. So a plus b will add up to 180. Or if you wanted to write in words, the sum of the angles in a linear pair is 180 degrees. These are just things that we assume.
And it's to some degree based on how we defined degrees, et cetera. Now you're saying, okay Sal, I get this postulates is just... These are just things that we assume to be true about the universe. But let's start building from there. Well, in order to build from there, we have to think about, what direction do we wanna build in?
And that's what conjectures do for us. They give us ideas of things that might be true. What do I mean by that? Well, let's say just from experience, you have a line like this. And you have a line like this. And you measure this angle. And you get to be 37 degrees.
And you measure this angle right over here. And you see that it is also 37 degrees. And you say, okay, is that a coincidence that you have these angles that are on either side of this intersection? This is often called vertical angles. Is that a coincidence it just happened to be the two that I chose?
But let's say that you do it with two different lines. So let's say you look at these two right over here. And you measure this angle is 80 degrees. And this one is also 80 degrees. So then you might start forming a conjecture right over here. Let me write this down.
The conjecture would be, all right, I keep seeing that vertical angles, every time I measure them, they seem to be the same. So maybe your conjecture is that vertical angles, vertical angles, are congruent, that they maybe always have the same measure. Now, a conjecture, you can't say that it's always true just because every time you measured, you saw it to be true.
So we have to now prove it. And if we prove it, then and only then will it become a theorem. So to go from a conjecture to a theorem... Lemme do it here, so it's on the video, or on the screen. To go from a conjecture to a theorem, you need a proof. So what's an example of a proof?
And you're gonna see a lot of proofs in geometry. In fact, that's a lot of the beauty of geometry. Let's say, let me just draw some vertical angles here. This is point A. This is point B. This is point C. This is point D. And let's call this one O. And what we want to do is prove that the measure that no matter which vertical angles we pick...
And so, these are just arbitrary ones. I'm saying it in a very general term, But we want to prove... Lemme write this, prove that the measure of angle AOD, so that's that one right over there, measure of angle AOD is equal to the measure of angle BOC, to BOC, right over there.
And if you're not interested, you could stop watching this video now. But if you wanna see a real proof, I'm about to do that. Well, we can say we know from the linear pair postulate that the measure of angle AOD plus the measure of... Lemme do it in a different color.
Plus the measure of angle AOB, plus the measure of angle AOB is going to be equal to 180 degrees. We also know that because BOC and BOA are also linear pair, we can make another statement. And actually, let me make it clear why we are saying that. The linear pair postulate can tell us that.
And the linear pair postulate can also tell us that the measure of angle BOC, BOC, plus the measure of angle AOB, plus the measure of angle AOB is also equal to 180 degrees. 'Cause once again, these are on the same... These are in a linear pair, so to speak. So this is interesting here.
What we could do is we could subtract the measure of angle AOB from both sides. So from that first one, we would get the measure of angle AOD is equal to 180. 180 minus the measure of angle AOB. And we could take the second statement right over here and we could say the measure of angle BOC is equal to 180 minus the measure of angle AOB.
And then we could use this other postulate right over here, the transitive postulate, right over there, or transitive property of equality to then say, all right, if these two things are equal to the same quantity, they must be equal to each other. So the measure of angle AOD must be equal to the measure of angle BOC.
And if you have two angles that no matter where they are, they have the same measure, they are going to be... Well, I just proved what I wanna show that here, that you have the same measure. And this is a general. So if you just have any two angles that are vertical, that are on opposite sides of an intersection like this, exactly opposite like that, then they're going to have the same measure.
So now, this conjecture that vertical angles are congruent, I can now promote it to being a theorem. It's something that I know is true. And now, I can build off of that.
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