WEBVTT mathematics/geometry/pyo 00:00:00.000 --> 00:00:02.200 Welcome back to Educator.com. 00:00:02.200 --> 00:00:13.300 The next lesson is on indirect proofs, and we are going to go over some theories on inequalities. 00:00:13.300 --> 00:00:19.200 An indirect proof: now, all of the other proofs that we have done until now, like the two-column proof, 00:00:19.200 --> 00:00:29.000 the paragraph proof, and the flow proofs--those are all direct proofs; those are all starting from point A, going directly to point B, 00:00:29.000 --> 00:00:33.700 your given statement down to the "prove" statement--those are all direct. 00:00:33.700 --> 00:00:42.400 Now, this one is an indirect proof, which means that you are going to still prove your statement; 00:00:42.400 --> 00:00:50.400 you are going to still prove something, but in an indirect way. 00:00:50.400 --> 00:00:59.900 To write an indirect proof, there are three steps, and the first step is this one right here. 00:00:59.900 --> 00:01:05.900 The first step is to assume that the conclusion is false--whatever conclusion you have, 00:01:05.900 --> 00:01:20.700 whatever statement you have, you are either going to state it as false, or you are going to state the opposite. 00:01:20.700 --> 00:01:24.600 That is all you are going to do: just state the opposite. 00:01:24.600 --> 00:01:31.400 By stating the opposite, you are going to try to prove that opposite statement. 00:01:31.400 --> 00:01:40.600 And as you prove the opposite statement, you are going to come across a contradiction of something that you know to be true. 00:01:40.600 --> 00:01:51.600 So, it could be some kind of fact; it could be a theorem, a postulate...something that we have learned until now. 00:01:51.600 --> 00:01:56.400 It is going to contradict; you are going to try to prove the opposite statement, 00:01:56.400 --> 00:02:07.900 and so you are basically proving that the opposite statement is false, and therefore, that the original statement is true. 00:02:07.900 --> 00:02:15.400 Instead of, like all of the proofs that we have done so far, proving that the original statement is true, 00:02:15.400 --> 00:02:20.800 you are proving that the opposite of the original is false, and therefore, that the original statement is true. 00:02:20.800 --> 00:02:25.500 So, that is what an indirect proof is. 00:02:25.500 --> 00:02:32.900 Again, step 1: You state the opposite of the statement; that is it. 00:02:32.900 --> 00:02:43.000 Then, step 2: You are going to actually write a reason, trying to prove that the opposite is true. 00:02:43.000 --> 00:02:56.500 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; 00:02:56.500 --> 00:03:03.500 what is involved in that opposite statement; and then eventually, you are going to come to a contradiction, 00:03:03.500 --> 00:03:13.500 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. 00:03:13.500 --> 00:03:32.600 Then, you say that, since that statement leads to a contradiction, the conclusion must be true. 00:03:32.600 --> 00:03:45.600 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." 00:03:45.600 --> 00:03:48.900 You are stating the opposite, that it is not sunny outside. 00:03:48.900 --> 00:04:07.200 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 00:04:07.200 --> 00:04:18.400 that it is a sunny day"; therefore, it leads to a contradiction, because you stated that it is not sunny. 00:04:18.400 --> 00:04:33.300 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." 00:04:33.300 --> 00:04:41.900 Now, I know that this is a really tough section; so we will try to go through step-by-step, 00:04:41.900 --> 00:04:53.000 and we will try to get you familiar, or a little more comfortable, with understanding indirect proofs. 00:04:53.000 --> 00:04:58.900 For these problems right here, we are actually just going to state the first step; we are going to start on the first step. 00:04:58.900 --> 00:05:02.500 And then, we will work our way through. 00:05:02.500 --> 00:05:07.100 State the assumption you would make to start an indirect proof. 00:05:07.100 --> 00:05:21.800 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. 00:05:21.800 --> 00:05:37.000 So, assume that...2 + 6 = 8, so assume that 2 + 6 does not equal 8. 00:05:37.000 --> 00:05:42.300 That is the step 1: assume that it is not equal to 8. 00:05:42.300 --> 00:06:01.400 Then, for number 2, your step 1 here: Assume the suspect is not guilty. 00:06:01.400 --> 00:06:08.300 That would be the assumption for that one. 00:06:08.300 --> 00:06:20.700 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, 00:06:20.700 --> 00:06:31.600 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, 00:06:31.600 --> 00:06:34.600 because the measure of angle A is smaller than the measure of angle B. 00:06:34.600 --> 00:06:41.400 So then, I could say it is 70; I could say that it is 61; I can say 100. 00:06:41.400 --> 00:06:53.600 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. 00:06:53.600 --> 00:07:07.600 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. 00:07:07.600 --> 00:07:19.700 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. 00:07:19.700 --> 00:07:27.700 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. 00:07:27.700 --> 00:07:35.400 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. 00:07:35.400 --> 00:07:48.100 Then, the measure of angle A is greater than or equal to the measure of angle B. 00:07:48.100 --> 00:07:58.000 OK, let's go over some inequality stuff; now, this is the definition of inequality: For any real numbers A and B, 00:07:58.000 --> 00:08:08.500 A is greater than B if and only if there is a positive number C such that A = B + C. 00:08:08.500 --> 00:08:27.100 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. 00:08:27.100 --> 00:08:44.200 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. 00:08:44.200 --> 00:08:48.500 Well, 10 is greater than B, so 10 is greater than 6. 00:08:48.500 --> 00:08:56.000 And there is a positive number C such that A = B + C. 00:08:56.000 --> 00:09:07.800 So, A (10) = B (6) plus C (which is 4); that means that C is 4. 00:09:07.800 --> 00:09:24.900 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. 00:09:24.900 --> 00:09:34.700 This has to be greater than this, because these two together make up 10. 00:09:34.700 --> 00:09:44.000 So, 10 by itself is going to be greater than 6, the B; that is what it is saying. 00:09:44.000 --> 00:09:51.800 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. 00:09:51.800 --> 00:09:55.500 A is going to be greater than B and C separately. 00:09:55.500 --> 00:09:59.000 First, let's go over some properties of inequality. 00:09:59.000 --> 00:10:06.300 The first one is the comparison property; with comparison, you know that you are comparing two things to each other. 00:10:06.300 --> 00:10:11.000 It could be two or more things, but you are just comparing things with each other--the comparison property. 00:10:11.000 --> 00:10:14.600 And then, of course, these are all properties of inequality. 00:10:14.600 --> 00:10:25.400 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; 00:10:25.400 --> 00:10:34.700 you can say A is equal to B--those are used to compare; you are just stating that one is compared to the other. 00:10:34.700 --> 00:10:41.300 The transitive property: now, we know the transitive property, but then, this is the transitive property of inequality. 00:10:41.300 --> 00:10:50.800 That just means, let's say, that if the measure of angle A is less than the measure of angle B, 00:10:50.800 --> 00:11:00.500 and the measure of angle B is less than the measure of angle C, then... 00:11:00.500 --> 00:11:10.400 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, 00:11:10.400 --> 00:11:13.300 and the measure of angle B is smaller than the measure of angle C... 00:11:13.300 --> 00:11:22.900 now, just to make this a little easier to see, I can write these angles (angles A, B, and C) 00:11:22.900 --> 00:11:29.200 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 00:11:29.200 --> 00:11:35.600 angle A small, and then I am going to write angle B bigger, like that. 00:11:35.600 --> 00:11:44.500 So, that shows that A is smaller than B; and then, B is less than C, so then C has to be even bigger. 00:11:44.500 --> 00:11:55.100 So then, what can we conclude there? Then the measure of angle A is less than the measure of angle C. 00:11:55.100 --> 00:12:02.400 That is the transitive property; it is just using the transitive property, but in the form of inequalities. 00:12:02.400 --> 00:12:27.100 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. 00:12:27.100 --> 00:12:35.500 This is the addition property, because you originally had A is greater than B; then you added C to both sides, 00:12:35.500 --> 00:12:39.300 and that is going to change the statement; it is still a true statement, 00:12:39.300 --> 00:12:46.500 but you are adding a number to each side, and that would be the addition property. 00:12:46.500 --> 00:12:51.900 We are really familiar with all of these properties; we all know the addition property and subtraction property. 00:12:51.900 --> 00:12:57.200 It is just adding something or subtracting something; and the same thing for multiplication and division. 00:12:57.200 --> 00:13:08.200 It is just multiplying something or dividing something; but you are using these in the form of inequalities. 00:13:08.200 --> 00:13:16.700 So then, the multiplication property would be like if you have, let's say, 5x > 20. 00:13:16.700 --> 00:13:32.200 Then, x is greater than 4; so that is a property of inequality, because 5x doesn't equal 20; 5x is greater than 20. 00:13:32.200 --> 00:13:40.600 And you are using these multiplication and division properties for inequalities. 00:13:40.600 --> 00:13:47.100 Again, with the comparison property, we are comparing two things to each other, one compared to the other. 00:13:47.100 --> 00:13:53.600 With the transitive property, if one is smaller than the other, and that is smaller than something else, 00:13:53.600 --> 00:14:01.800 then the original would be smaller than the measure of angle C. 00:14:01.800 --> 00:14:09.200 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. 00:14:09.200 --> 00:14:13.600 And the same happens for the multiplication and division properties. 00:14:13.600 --> 00:14:24.600 OK, so then, this exterior angle inequality theorem is from the inequality theorem that we went over before, 00:14:24.600 --> 00:14:46.400 where we said that, if A is greater than B, and then A equals B + C...how we talked about that: 00:14:46.400 --> 00:14:53.000 because this and this together add up to get A, that means that A is greater than B itself. 00:14:53.000 --> 00:15:01.200 So, in the same way, this exterior angle inequality theorem applies the same concept. 00:15:01.200 --> 00:15:15.500 If I have an exterior angle (and if you remember, the exterior angle theorem was when we had 00:15:15.500 --> 00:15:28.200 an exterior angle of a triangle--so then, if this is, let's say, angle 1, then the exterior angle equals 00:15:28.200 --> 00:15:34.100 the sum of its two remote interior angles), remember the two remote interior angles: 00:15:34.100 --> 00:15:43.000 there are three angles of a triangle; now, it is not this one that is next to it, that forms a linear pair; 00:15:43.000 --> 00:15:47.400 it is the other two angles that are far away from it, that are not touching it--remote interior angles. 00:15:47.400 --> 00:15:51.900 So, these two would be considered the remote interior angles. 00:15:51.900 --> 00:16:02.900 If this is A and B, then the measure of angle 1 equals the measure of angle A, plus the measure of angle B. 00:16:02.900 --> 00:16:10.400 This is the exterior angle theorem; now, the exterior angle inequality theorem says that, 00:16:10.400 --> 00:16:23.800 if the measure of angle 1 equals this plus this, then the measure of angle 1 is greater than the measure of angle A, 00:16:23.800 --> 00:16:29.800 and the measure of angle 1 is also greater than the measure of angle B, 00:16:29.800 --> 00:16:36.800 because again, these two together add up to be the same measure as angle 1. 00:16:36.800 --> 00:16:42.900 So therefore, angle 1 is greater than each of these by itself. 00:16:42.900 --> 00:16:47.300 And that is the exterior angle inequality theorem. 00:16:47.300 --> 00:16:52.100 This is the exterior angle theorem, and then this is the exterior angle inequality theorem. 00:16:52.100 --> 00:17:02.400 This has to do with the inequalities; and just to read it to you, "If an angle is an exterior angle of a triangle, 00:17:02.400 --> 00:17:12.300 then its measure is greater...this is supposed to be "than"...the measures of either of its remote interior angles." 00:17:12.300 --> 00:17:21.800 That means that it is either of its remote interior angles, this one and this one, just by itself. 00:17:21.800 --> 00:17:40.400 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; 00:17:40.400 --> 00:17:55.000 the measure of angle B is 45; I know that these two...100 = 55 (because the measure of angle 1 is 100, 00:17:55.000 --> 00:18:14.000 and the measure of angle A is 55) + 45; so A and B together equal 100. 00:18:14.000 --> 00:18:19.600 Then, since these together add up to be 100, and 100 is greater than each of these 00:18:19.600 --> 00:18:33.300 (100 is greater than 55, and then 100 is greater than 45), that is what this inequality theorem is saying. 00:18:33.300 --> 00:18:37.900 OK, let's go over our examples: Draw a diagram for the statement. 00:18:37.900 --> 00:18:49.100 Angle 1 is an exterior angle of a triangle greater than each of the remote interior angles, 2 and 3. 00:18:49.100 --> 00:18:56.900 So, basically, you have to draw the same diagram that we drew in the last slide. 00:18:56.900 --> 00:19:07.900 You can draw any sort of exterior angle; if I have a triangle like this, I can draw an exterior angle right here; 00:19:07.900 --> 00:19:14.800 so then, the exterior angle is angle 1; there is angle 1; and then, the remote interior angles, 00:19:14.800 --> 00:19:23.700 2 and 3, have to be away from this angle, in here and in here. 00:19:23.700 --> 00:19:31.900 You can draw it however you want--like this--as long as you draw it 1, 2, 3. 00:19:31.900 --> 00:19:42.200 And it is just so that you have some sort of visual, or you understand what it would look like, 00:19:42.200 --> 00:19:57.700 and understand that this is the exterior angle inequality theorem, and how this angle would be greater than each of these angles by itself. 00:19:57.700 --> 00:20:10.200 OK, Example 2: Name the property for each statement: If AB < CD, then AB + CD < CD + EF. 00:20:10.200 --> 00:20:28.500 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. 00:20:28.500 --> 00:20:34.600 The next one: if 6x is greater than 30, then x is greater than 5; so what happened here? 00:20:34.600 --> 00:20:46.900 I divided this, so this is the division property of inequality. 00:20:46.900 --> 00:20:55.400 If the measure of angle A is less than the measure of angle B (let me just draw this again), 00:20:55.400 --> 00:21:01.900 and the measure of angle C is less than the measure of angle A (that means that C is really small, like that), 00:21:01.900 --> 00:21:06.700 then the measure of angle C is less than the measure of angle B. 00:21:06.700 --> 00:21:23.100 So, this is, we know, the transitive property of inequality. 00:21:23.100 --> 00:21:47.900 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. 00:21:47.900 --> 00:21:51.500 And then, in the next example, we will actually do an indirect proof. 00:21:51.500 --> 00:21:55.600 The first one: Sally is sick today. 00:21:55.600 --> 00:22:16.700 Your first step is to say, "Assume that Sally is not sick today." 00:22:16.700 --> 00:22:24.300 If you were to continue this for steps 2 and 3 of an indirect proof, you can say something like... 00:22:24.300 --> 00:22:30.600 since this is what you are trying to prove, and you are going to end up proving this false... 00:22:30.600 --> 00:22:45.600 if you say, "If Sally is not sick today, then she would have come to school today; 00:22:45.600 --> 00:22:54.600 she doesn't sound so good; so since she didn't come today, therefore she must be sick today." 00:22:54.600 --> 00:23:04.100 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. 00:23:04.100 --> 00:23:10.400 And for your last statement, you can say, "Therefore, Sally must be sick today," or something like that. 00:23:10.400 --> 00:23:15.900 The exterior angle of a triangle has a larger angle measure than one of its remote interior angles. 00:23:15.900 --> 00:23:58.600 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) 00:23:58.600 --> 00:24:18.200 "angle measure than one of its remote interior angles." 00:24:18.200 --> 00:24:27.400 The difference between what is written here, this statement, and the step 1, is to say that it does not. 00:24:27.400 --> 00:24:34.800 And the next one: 41 is not divisible by 3; now again, you are stating the opposite. 00:24:34.800 --> 00:24:55.100 You say, "Assume that 41 is divisible by 3." 00:24:55.100 --> 00:25:07.100 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." 00:25:07.100 --> 00:25:23.700 "When I divide it, there is a remainder of 2; therefore, this statement is false," or "it contradicts what we know is true." 00:25:23.700 --> 00:25:30.900 And then, in step 3, you can say, "41 must not be divisible by 3"; then you are stating that this one is true. 00:25:30.900 --> 00:25:39.600 Just like a direct proof, in the end, you are going to eventually state that it is true. 00:25:39.600 --> 00:25:53.900 And just to explain indirect proofs again: all of the other proofs that we have done, the direct proofs, 00:25:53.900 --> 00:26:07.600 are just going straight from the given to the "prove" statement; you are just directly proving from the first step to the last step. 00:26:07.600 --> 00:26:10.600 It is just that you are proving it. 00:26:10.600 --> 00:26:16.100 With an indirect proof, you are going to actually state that the given statement is the opposite. 00:26:16.100 --> 00:26:25.900 You are actually proving that the opposite of the original statement is false, and therefore proving that it is true. 00:26:25.900 --> 00:26:32.500 Instead of proving that the original statement is true, you are going to prove that the opposite of the original is false. 00:26:32.500 --> 00:26:37.000 So, it is an indirect way of writing these proofs. 00:26:37.000 --> 00:26:44.500 Then here, in this problem, we are actually going to write an indirect proof; so let's take a look at this. 00:26:44.500 --> 00:26:50.900 The given is that lines p and q are not parallel. 00:26:50.900 --> 00:26:57.600 And then, we are going to prove that the measure of angle 1 is not equal to the measure of angle 2. 00:26:57.600 --> 00:27:01.600 Now, you can do a regular proof with this; you can do a two-column proof with this. 00:27:01.600 --> 00:27:07.600 But this is just another type. 00:27:07.600 --> 00:27:10.000 To do an indirect proof, remember, we are going to do three steps. 00:27:10.000 --> 00:27:20.000 Step 1: we are going to say that the given is false, or its opposite. 00:27:20.000 --> 00:27:32.000 We are going to say, "Lines p and q are parallel." 00:27:32.000 --> 00:27:41.000 And then, you can also write (and it is probably better), "Assume that lines p and q are parallel." 00:27:41.000 --> 00:27:50.500 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. 00:27:50.500 --> 00:28:02.700 Now, these are alternate interior angles; if alternate interior angles are congruent, then we know that these lines have to be parallel. 00:28:02.700 --> 00:28:30.700 So, I can say, "Angles 1 and 2 are alternate interior angles; if alternate interior..." 00:28:30.700 --> 00:29:32.100 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." 00:29:32.100 --> 00:29:40.200 "However"...now, since we said that angles 1 and 2 are alternate interior angles, and that, 00:29:40.200 --> 00:29:50.400 if two lines are cut by a transversal so that alternate interior angles are congruent, then the two lines have to be parallel, 00:29:50.400 --> 00:30:13.100 well, doesn't this contradict the given statement?...so, "However, this contradicts the given statement." 00:30:13.100 --> 00:30:45.100 So, for my step 3, I am going to say, "Therefore, since the assumption" right here in step 1 "leads to a contradiction, 00:30:45.100 --> 00:31:13.400 the assumption" again, step 1 "must be false, and therefore, the measure of angle 1 does not equal the measure of angle 2." 00:31:13.400 --> 00:31:19.600 If you were to write your own indirect proof, yours might be a little bit different than mine. 00:31:19.600 --> 00:31:29.100 All you have to make sure you include is that, since we are assuming that lines p and q are parallel, 00:31:29.100 --> 00:31:36.900 angles 1 and 2, since they are remote interior angles, must be congruent in order for the lines to be parallel. 00:31:36.900 --> 00:31:41.200 If the lines are not parallel, that is fine, but the angles will not be congruent. 00:31:41.200 --> 00:31:50.900 If the two lines are cut by a transversal so that alternate interior angles are congruent, then the two lines are parallel. 00:31:50.900 --> 00:31:56.700 They must be parallel; but that contradicts the given statement, which is that they are not parallel. 00:31:56.700 --> 00:32:10.100 So, since this assumption right here, that they are parallel, is a contradiction--it contradicts what we know to be true, 00:32:10.100 --> 00:32:29.900 this theorem--then the assumption is false; you have to include that this is false. Therefore, this is true. 00:32:29.900 --> 00:32:47.500 We are explaining it in a way where it contradicts, and then saying, "Therefore, since it is a contradiction, the assumption is false." 00:32:47.500 --> 00:32:57.800 The measure of angle 1 is not equal to the measure of angle 2. 00:32:57.800 --> 00:33:02.000 And that is it for this lesson. 00:33:02.000 --> 00:33:13.800 Indirect proofs are a little bit challenging, but again, all you have to do is just show that this contradicts the given statement. 00:33:13.800 --> 00:33:26.800 And once you do that, you can say that, since it does contradict, the assumption is false; therefore, this has to be true. 00:33:26.800 --> 00:33:30.000 Well, that is it for this lesson; thank you for watching Educator.com.