WEBVTT mathematics/geometry/pyo
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Welcome back to Educator.com.
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The next lesson is on indirect proofs, and we are going to go over some theories on inequalities.
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An indirect proof: now, all of the other proofs that we have done until now, like the two-column proof,
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the paragraph proof, and the flow proofs--those are all direct proofs; those are all starting from point A, going directly to point B,
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your given statement down to the "prove" statement--those are all direct.
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Now, this one is an indirect proof, which means that you are going to still prove your statement;
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you are going to still prove something, but in an indirect way.
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To write an indirect proof, there are three steps, and the first step is this one right here.
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The first step is to assume that the conclusion is false--whatever conclusion you have,
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whatever statement you have, you are either going to state it as false, or you are going to state the opposite.
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That is all you are going to do: just state the opposite.
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By stating the opposite, you are going to try to prove that opposite statement.
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And as you prove the opposite statement, you are going to come across a contradiction of something that you know to be true.
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So, it could be some kind of fact; it could be a theorem, a postulate...something that we have learned until now.
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It is going to contradict; you are going to try to prove the opposite statement,
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and so you are basically proving that the opposite statement is false, and therefore, that the original statement is true.
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Instead of, like all of the proofs that we have done so far, proving that the original statement is true,
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you are proving that the opposite of the original is false, and therefore, that the original statement is true.
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So, that is what an indirect proof is.
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Again, step 1: You state the opposite of the statement; that is it.
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Then, step 2: You are going to actually write a reason, trying to prove that the opposite is true.
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But you won't be able to; this is more like your reasoning--you are showing your steps and your reasons behind why it is true;
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what is involved in that opposite statement; and then eventually, you are going to come to a contradiction,
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because obviously, you are showing that the opposite is false, so it is going to come to a contradiction of some known fact or something.
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Then, you say that, since that statement leads to a contradiction, the conclusion must be true.
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For example, if I say, "It is sunny outside," if that is my statement, then your step 1 is going to say, "It is not sunny outside."
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You are stating the opposite, that it is not sunny outside.
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And then, you can say, for step 2, "Well, it is bright outside; the sun is out; it is shining; it is hot; therefore, all of these show
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that it is a sunny day"; therefore, it leads to a contradiction, because you stated that it is not sunny.
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But then, everything that you are saying proves it being sunny; so then, you say, "Therefore, the conclusion must be true--it must be sunny outside."
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Now, I know that this is a really tough section; so we will try to go through step-by-step,
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and we will try to get you familiar, or a little more comfortable, with understanding indirect proofs.
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For these problems right here, we are actually just going to state the first step; we are going to start on the first step.
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And then, we will work our way through.
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State the assumption you would make to start an indirect proof.
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My step 1 for this one (all we are going to do is just the step 1): remember: step 1 is to assume that the opposite is true.
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So, assume that...2 + 6 = 8, so assume that 2 + 6 does not equal 8.
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That is the step 1: assume that it is not equal to 8.
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Then, for number 2, your step 1 here: Assume the suspect is not guilty.
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That would be the assumption for that one.
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And then, step 1 for this: this is actually a little bit different--if you say that the measure of angle A is less than the measure of angle B,
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you can say...let's say, if the measure of angle A is 60, then that means that the measure of angle B would be anything that is greater,
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because the measure of angle A is smaller than the measure of angle B.
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So then, I could say it is 70; I could say that it is 61; I can say 100.
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If I am stating the opposite, that means that I am saying that the measure of angle A is greater than the measure of angle B.
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But also, because the measure of angle A can only be smaller, that means the measure of angle A and the measure of angle B cannot be the same.
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That means that, if the measure of angle A is 60, the measure of angle B has to be greater; that means that the measure of angle B cannot be 60.
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If it is the opposite, then you have to say that the measure of angle A could be greater than the measure of angle B, and it could be equal.
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Because this one doesn't say that it can be equal, for this one you can say that it can be equal, because it is the opposite.
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Then, the measure of angle A is greater than or equal to the measure of angle B.
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OK, let's go over some inequality stuff; now, this is the definition of inequality: For any real numbers A and B,
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A is greater than B if and only if there is a positive number C such that A = B + C.
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OK, just this part right here, "if and only if": remember, don't get confused by that; it just means that this statement and the converse are true.
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Now, we have two numbers, A and B; A is greater than B, so let's say A is 10 (just as an example), and B is, let's say, 6.
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Well, 10 is greater than B, so 10 is greater than 6.
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And there is a positive number C such that A = B + C.
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So, A (10) = B (6) plus C (which is 4); that means that C is 4.
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And it is just saying that if there are two numbers, B and C, that add up together to get A, then A must be greater than B.
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This has to be greater than this, because these two together make up 10.
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So, 10 by itself is going to be greater than 6, the B; that is what it is saying.
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You can also say that 10 is going to be greater than C: 10 is going to be greater than B, and 10 is going to be greater than C.
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A is going to be greater than B and C separately.
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First, let's go over some properties of inequality.
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The first one is the comparison property; with comparison, you know that you are comparing two things to each other.
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It could be two or more things, but you are just comparing things with each other--the comparison property.
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And then, of course, these are all properties of inequality.
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So, when you are comparing, let's say, A and B, that means that A is greater than B, or you can say A is less than B;
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you can say A is equal to B--those are used to compare; you are just stating that one is compared to the other.
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The transitive property: now, we know the transitive property, but then, this is the transitive property of inequality.
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That just means, let's say, that if the measure of angle A is less than the measure of angle B,
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and the measure of angle B is less than the measure of angle C, then...
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and to see what we are going to write as our conclusion, if the measure of angle A is smaller than the measure of angle B,
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and the measure of angle B is smaller than the measure of angle C...
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now, just to make this a little easier to see, I can write these angles (angles A, B, and C)
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according to size: so let's say, since the measure of angle A is smaller than the measure of angle B, that I am going to write
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angle A small, and then I am going to write angle B bigger, like that.
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So, that shows that A is smaller than B; and then, B is less than C, so then C has to be even bigger.
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So then, what can we conclude there? Then the measure of angle A is less than the measure of angle C.
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That is the transitive property; it is just using the transitive property, but in the form of inequalities.
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Addition and subtraction properties: now, for this one right here, if I say that A is greater than B, then A + C is greater than B + C.
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This is the addition property, because you originally had A is greater than B; then you added C to both sides,
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and that is going to change the statement; it is still a true statement,
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but you are adding a number to each side, and that would be the addition property.
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We are really familiar with all of these properties; we all know the addition property and subtraction property.
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It is just adding something or subtracting something; and the same thing for multiplication and division.
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It is just multiplying something or dividing something; but you are using these in the form of inequalities.
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So then, the multiplication property would be like if you have, let's say, 5x > 20.
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Then, x is greater than 4; so that is a property of inequality, because 5x doesn't equal 20; 5x is greater than 20.
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And you are using these multiplication and division properties for inequalities.
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Again, with the comparison property, we are comparing two things to each other, one compared to the other.
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With the transitive property, if one is smaller than the other, and that is smaller than something else,
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then the original would be smaller than the measure of angle C.
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The addition and subtraction properties are the same thing as the addition and subtraction properties of equality; it is just that you are using inequality.
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And the same happens for the multiplication and division properties.
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OK, so then, this exterior angle inequality theorem is from the inequality theorem that we went over before,
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where we said that, if A is greater than B, and then A equals B + C...how we talked about that:
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because this and this together add up to get A, that means that A is greater than B itself.
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So, in the same way, this exterior angle inequality theorem applies the same concept.
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If I have an exterior angle (and if you remember, the exterior angle theorem was when we had
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an exterior angle of a triangle--so then, if this is, let's say, angle 1, then the exterior angle equals
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the sum of its two remote interior angles), remember the two remote interior angles:
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there are three angles of a triangle; now, it is not this one that is next to it, that forms a linear pair;
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it is the other two angles that are far away from it, that are not touching it--remote interior angles.
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So, these two would be considered the remote interior angles.
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If this is A and B, then the measure of angle 1 equals the measure of angle A, plus the measure of angle B.
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This is the exterior angle theorem; now, the exterior angle inequality theorem says that,
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if the measure of angle 1 equals this plus this, then the measure of angle 1 is greater than the measure of angle A,
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and the measure of angle 1 is also greater than the measure of angle B,
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because again, these two together add up to be the same measure as angle 1.
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So therefore, angle 1 is greater than each of these by itself.
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And that is the exterior angle inequality theorem.
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This is the exterior angle theorem, and then this is the exterior angle inequality theorem.
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This has to do with the inequalities; and just to read it to you, "If an angle is an exterior angle of a triangle,
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then its measure is greater...this is supposed to be "than"...the measures of either of its remote interior angles."
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That means that it is either of its remote interior angles, this one and this one, just by itself.
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And then, an example would be...let's say the measure of angle 1 is 100; the measure of angle A is, let's say, 55;
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the measure of angle B is 45; I know that these two...100 = 55 (because the measure of angle 1 is 100,
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and the measure of angle A is 55) + 45; so A and B together equal 100.
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Then, since these together add up to be 100, and 100 is greater than each of these
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(100 is greater than 55, and then 100 is greater than 45), that is what this inequality theorem is saying.
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OK, let's go over our examples: Draw a diagram for the statement.
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Angle 1 is an exterior angle of a triangle greater than each of the remote interior angles, 2 and 3.
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So, basically, you have to draw the same diagram that we drew in the last slide.
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You can draw any sort of exterior angle; if I have a triangle like this, I can draw an exterior angle right here;
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so then, the exterior angle is angle 1; there is angle 1; and then, the remote interior angles,
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2 and 3, have to be away from this angle, in here and in here.
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You can draw it however you want--like this--as long as you draw it 1, 2, 3.
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And it is just so that you have some sort of visual, or you understand what it would look like,
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and understand that this is the exterior angle inequality theorem, and how this angle would be greater than each of these angles by itself.
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OK, Example 2: Name the property for each statement: If AB < CD, then AB + CD < CD + EF.
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So then, all we did here is, from the hypothesis statement to the conclusion statement, added the EF's; so this is the addition property of inequality.
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The next one: if 6x is greater than 30, then x is greater than 5; so what happened here?
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I divided this, so this is the division property of inequality.
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If the measure of angle A is less than the measure of angle B (let me just draw this again),
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and the measure of angle C is less than the measure of angle A (that means that C is really small, like that),
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then the measure of angle C is less than the measure of angle B.
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So, this is, we know, the transitive property of inequality.
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Let's go back to our indirect proofs part of the section; and we are going to work on a few more step 1's.
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And then, in the next example, we will actually do an indirect proof.
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The first one: Sally is sick today.
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Your first step is to say, "Assume that Sally is not sick today."
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If you were to continue this for steps 2 and 3 of an indirect proof, you can say something like...
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since this is what you are trying to prove, and you are going to end up proving this false...
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if you say, "If Sally is not sick today, then she would have come to school today;
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she doesn't sound so good; so since she didn't come today, therefore she must be sick today."
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And you can say something about how it contradicts the statement, so Sally's not being sick today is not true, because she is sick.
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And for your last statement, you can say, "Therefore, Sally must be sick today," or something like that.
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The exterior angle of a triangle has a larger angle measure than one of its remote interior angles.
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Then, for this one, you would say, "Assume that the exterior angle of a triangle does not have a larger" (that is how you would say it)
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"angle measure than one of its remote interior angles."
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The difference between what is written here, this statement, and the step 1, is to say that it does not.
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And the next one: 41 is not divisible by 3; now again, you are stating the opposite.
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You say, "Assume that 41 is divisible by 3."
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And for this last one, in your step 2, in your reason, you could say, "Well, if 41 is divisible by 3, then there should not be a remainder."
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"When I divide it, there is a remainder of 2; therefore, this statement is false," or "it contradicts what we know is true."
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And then, in step 3, you can say, "41 must not be divisible by 3"; then you are stating that this one is true.
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Just like a direct proof, in the end, you are going to eventually state that it is true.
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And just to explain indirect proofs again: all of the other proofs that we have done, the direct proofs,
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are just going straight from the given to the "prove" statement; you are just directly proving from the first step to the last step.
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It is just that you are proving it.
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With an indirect proof, you are going to actually state that the given statement is the opposite.
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You are actually proving that the opposite of the original statement is false, and therefore proving that it is true.
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Instead of proving that the original statement is true, you are going to prove that the opposite of the original is false.
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So, it is an indirect way of writing these proofs.
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Then here, in this problem, we are actually going to write an indirect proof; so let's take a look at this.
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The given is that lines *p* and *q* are not parallel.
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And then, we are going to prove that the measure of angle 1 is not equal to the measure of angle 2.
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Now, you can do a regular proof with this; you can do a two-column proof with this.
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But this is just another type.
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To do an indirect proof, remember, we are going to do three steps.
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Step 1: we are going to say that the given is false, or its opposite.
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We are going to say, "Lines *p* and *q* are parallel."
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And then, you can also write (and it is probably better), "Assume that lines *p* and *q* are parallel."
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My step 2: I am going to start...remember: I am trying to say that the measure of angle 1 is not congruent to the measure of angle 2.
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Now, these are alternate interior angles; if alternate interior angles are congruent, then we know that these lines have to be parallel.
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So, I can say, "Angles 1 and 2 are alternate interior angles; if alternate interior..."
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Or, let's say, "If two lines are cut by a transversal so that alternate interior angles are congruent, then the two lines are parallel."
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"However"...now, since we said that angles 1 and 2 are alternate interior angles, and that,
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if two lines are cut by a transversal so that alternate interior angles are congruent, then the two lines have to be parallel,
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well, doesn't this contradict the given statement?...so, "However, this contradicts the given statement."
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So, for my step 3, I am going to say, "Therefore, since the assumption" right here in step 1 "leads to a contradiction,
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the assumption" again, step 1 "must be false, and therefore, the measure of angle 1 does not equal the measure of angle 2."
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If you were to write your own indirect proof, yours might be a little bit different than mine.
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All you have to make sure you include is that, since we are assuming that lines *p* and *q* are parallel,
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angles 1 and 2, since they are remote interior angles, must be congruent in order for the lines to be parallel.
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If the lines are not parallel, that is fine, but the angles will not be congruent.
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If the two lines are cut by a transversal so that alternate interior angles are congruent, then the two lines are parallel.
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They must be parallel; but that contradicts the given statement, which is that they are not parallel.
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So, since this assumption right here, that they are parallel, is a contradiction--it contradicts what we know to be true,
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this theorem--then the assumption is false; you have to include that this is false. Therefore, this is true.
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We are explaining it in a way where it contradicts, and then saying, "Therefore, since it is a contradiction, the assumption is false."
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The measure of angle 1 is not equal to the measure of angle 2.
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And that is it for this lesson.
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Indirect proofs are a little bit challenging, but again, all you have to do is just show that this contradicts the given statement.
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And once you do that, you can say that, since it does contradict, the assumption is false; therefore, this has to be true.
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Well, that is it for this lesson; thank you for watching Educator.com.