WEBVTT mathematics/geometry/pyo 00:00:00.000 --> 00:00:02.100 Welcome back to Educator.com. 00:00:02.100 --> 00:00:06.600 This next lesson is on conditional statements. 00:00:06.600 --> 00:00:14.200 If/then statements are called conditional statements, or conditionals. 00:00:14.200 --> 00:00:25.000 When you have a statement in the form of if something, then something else, then that is considered a conditional statement. 00:00:25.000 --> 00:00:33.900 If you have a statement "I use an umbrella when it rains," you can rewrite it as a conditional in if/then form. 00:00:33.900 --> 00:00:44.400 So, "If it is raining, then I use an umbrella": that would be the conditional of the statement "I use an umbrella when it rains." 00:00:44.400 --> 00:00:51.600 When do you use an umbrella? When it rains, right? So, "If it is raining, then I use an umbrella." 00:00:51.600 --> 00:00:57.400 And that would be considered a conditional statement. 00:00:57.400 --> 00:01:12.100 If...this part right here, "If it is raining"--the phrase after the "if" is called the hypothesis. 00:01:12.100 --> 00:01:20.000 And then, the statement after the "then" is called the conclusion. 00:01:20.000 --> 00:01:26.800 If it is raining, then I use an umbrella: this part right here is known as the hypothesis; "then I use an umbrella"--that is the conclusion. 00:01:26.800 --> 00:01:31.600 That is what is going to result from the hypothesis. 00:01:31.600 --> 00:01:44.700 You can also think of the hypothesis as p; p is the hypothesis, and q is the conclusion. 00:01:44.700 --> 00:01:58.600 You can write this as a statement if p, then q, because p is the hypothesis; so it is if the hypothesis, then the conclusion. 00:01:58.600 --> 00:02:14.000 And as symbols, you can write it like this: p → q; p implies q, and that would be the symbol for this condition, "if p, then q." 00:02:14.000 --> 00:02:19.900 Again, the statement after the "if" is the hypothesis; the statement after the "then" is the conclusion. 00:02:19.900 --> 00:02:31.400 And then, it is if p, then q; you can also denote it as this, p → q; and that is "p implies q." 00:02:31.400 --> 00:02:37.900 Now, you could write this in a couple of different ways; you don't always have to write it "if" and "then." 00:02:37.900 --> 00:02:45.300 And it is still going to be considered a conditional: back to this example, "If it is raining, then I use an umbrella." 00:02:45.300 --> 00:02:57.400 If you write it without the "then," here is "then": If it is raining, I will use an umbrella; you can write it like that, too. 00:02:57.400 --> 00:03:04.000 "If it is raining, then I use an umbrella" can also be "If it is raining, I will just use an umbrella." 00:03:04.000 --> 00:03:12.500 You can also write it using "when" instead of the "if"; you are going to use the word when instead of if. 00:03:12.500 --> 00:03:19.300 "When it is raining, then I use an umbrella": just because you don't see an if there... 00:03:19.300 --> 00:03:24.800 this is still going to be the hypothesis, and then this is the conclusion. 00:03:24.800 --> 00:03:33.200 You can also reword it by stating the hypothesis at the end of it: "I use an umbrella if it is raining." 00:03:33.200 --> 00:03:42.400 Remember to always look for that word "if": I use an umbrella if it is raining, or I use an umbrella when it is raining. 00:03:42.400 --> 00:03:54.500 Just keep that in mind: the hypothesis doesn't always have to be in the front. 00:03:54.500 --> 00:04:07.500 Let's identify the hypothesis and the conclusion: the first one: I am going to make the hypothesis red, and the conclusion will be... 00:04:07.500 --> 00:04:16.800 If it is Tuesday, then Phil plays tennis: well, the hypothesis, I know, is "if it is Tuesday." 00:04:16.800 --> 00:04:24.100 So, "it is Tuesday" will be the hypothesis; then what is going to happen as a result? 00:04:24.100 --> 00:04:28.400 Phil is going to play tennis; that is the conclusion. 00:04:28.400 --> 00:04:31.300 If it is Tuesday, then Phil plays tennis. 00:04:31.300 --> 00:04:40.100 The next one: Three points that lie on a line are collinear. 00:04:40.100 --> 00:04:50.200 Now, this is not written as a conditional statement; so let's rewrite this in if/then form. 00:04:50.200 --> 00:05:19.300 Three points that lie on a line are collinear; If three points lie on a line, then they are collinear. 00:05:19.300 --> 00:05:31.000 My hypothesis, then, is "three points lie on a line"; and then, my conclusion is going to be "then they are collinear." 00:05:31.000 --> 00:05:39.100 Now, notice how, when I identify the hypothesis and conclusion, I am not including the "if" and the "then"; it is following the if and following the then. 00:05:39.100 --> 00:05:49.700 The next one: You are at least 21 years old if you are an adult. 00:05:49.700 --> 00:05:58.800 If you look at this, I see an "if" right here; so "you are at least 21 years old," if "you are an adult." 00:05:58.800 --> 00:06:07.700 Right here, "if you are an adult"--that is going to be the hypothesis; this is an example of when the hypothesis is written at the end of the statement. 00:06:07.700 --> 00:06:19.100 If you are an adult, then you are at least 21 years old. 00:06:19.100 --> 00:06:33.600 These examples, we are going to write in if/then form; adjacent angles have a common vertex. 00:06:33.600 --> 00:06:58.100 If angles are adjacent, then they have a common vertex. 00:06:58.100 --> 00:07:04.300 The next one: Glass objects are fragile; what is fragile?--glass objects. 00:07:04.300 --> 00:07:13.600 So, if the objects...you can write this a couple of different ways. 00:07:13.600 --> 00:07:23.500 You can say, "If the objects are made of glass"; you can say, "If these objects are glass objects..." 00:07:23.500 --> 00:07:39.500 I am just going to say, "If the objects are glass, then..." what?..."they are fragile." 00:07:39.500 --> 00:07:54.000 And the third one: An angle is obtuse if its measure is greater than 90 degrees. 00:07:54.000 --> 00:08:42.100 If..."its"...we want to rewrite this word; if an angle measures greater than 90 degrees, then it is obtuse. 00:08:42.100 --> 00:08:56.700 OK, when we are given a conditional, we can write those given statements in three other forms, 00:08:56.700 --> 00:09:03.200 meaning that we can change the conditionals around in three different ways. 00:09:03.200 --> 00:09:09.800 And the first way is the converse way: converse statements. 00:09:09.800 --> 00:09:13.800 Oh, and then, we are going to go over each of these separately; so converse statements is the first one, 00:09:13.800 --> 00:09:19.400 then inverse statements, then contrapositive statements; so just keep in mind that there are three different ways. 00:09:19.400 --> 00:09:26.000 And the first one, converse statements, is when you interchange the hypothesis and the conclusion. 00:09:26.000 --> 00:09:31.000 So, remember how we had if p, then q; the hypothesis is p; the conclusion is q. 00:09:31.000 --> 00:09:42.700 When you switch the p and the q, that is a converse; so what happens then is: it becomes if q, then p. 00:09:42.700 --> 00:09:53.000 The if and then are still the same; you are still writing the conditional; but you are just switching the hypothesis and the conclusion. 00:09:53.000 --> 00:09:57.700 And when you write the converse, it doesn't necessarily have to be true. 00:09:57.700 --> 00:10:04.900 It can be true or false; so again, this is going to be if q, then p. 00:10:04.900 --> 00:10:18.200 And remember: our conditional statements were p to q, but then the converse is going to be q to p, q implies p, because we are switching them. 00:10:18.200 --> 00:10:24.800 Here is an example: If it is raining, then I use an umbrella--that is the given conditional statement. 00:10:24.800 --> 00:10:32.500 Then, the converse, by switching: this is the hypothesis; "then I use an umbrella"--that is the conclusion. 00:10:32.500 --> 00:10:42.800 You are going to interchange these two; so then, "If I use an umbrella, then it is raining." 00:10:42.800 --> 00:10:49.800 This is the converse statement, because you switched the hypothesis and the conclusion. 00:10:49.800 --> 00:11:05.200 This is p; this is q; so then, this became q, and this is p; the converse just interchanges them. 00:11:05.200 --> 00:11:10.400 Now, remember from the last section: we went over counter-examples. 00:11:10.400 --> 00:11:23.300 Whenever you have some given statement, and you need to prove that it is false, then you give an example of when that statement is not true. 00:11:23.300 --> 00:11:28.700 And that is when you can prove that it is false. 00:11:28.700 --> 00:11:33.900 And like I said earlier, converse statements are not necessarily true; they are going to be true or false. 00:11:33.900 --> 00:11:44.300 If it is true, then you can leave it at that; but if it is false, then you need to give a counter-example--an example of why it is false, or when it is false. 00:11:44.300 --> 00:11:52.500 Write the converse of each given statement; decide if it is true or false; if false, write a counter-example. 00:11:52.500 --> 00:11:57.900 This one: Adjacent angles have a common side. 00:11:57.900 --> 00:12:02.400 Now, that is the given statement; we need to find the converse statement. 00:12:02.400 --> 00:12:34.500 So, if you want to write this as a conditional (meaning an if/then statement), then you say, "If angles are adjacent, then they have a common side." 00:12:34.500 --> 00:12:47.500 Then, the converse is going to be, "If"...now remember: again, you are not putting "then" first; 00:12:47.500 --> 00:12:52.800 you are keeping the "if" and the "then" statement, but you are just interchanging these two; 00:12:52.800 --> 00:13:17.100 so, "if angles have a common side, then they are adjacent." 00:13:17.100 --> 00:13:25.500 Now, we know that this statement right here is true: "If angles are adjacent, then they have a common side"; that is true. 00:13:25.500 --> 00:13:44.300 "If angles have a common side, then they are adjacent": well, if I have an angle like this; this is A...angle ABC, D... 00:13:44.300 --> 00:14:02.100 this is angle 1; this is angle 2; now, I know that angles 1 and 2 are adjacent angles, and they have a common side; 00:14:02.100 --> 00:14:06.700 that is the statement right here, and it is true. 00:14:06.700 --> 00:14:11.100 Now, if angles have a common side, then does that make them adjacent? 00:14:11.100 --> 00:14:32.000 Well, let's look at this: I see angle 2 right here, this angle, with this angle; angles 2 and ABC have a common side, which is this right here. 00:14:32.000 --> 00:14:37.600 This is their common side; but they are not adjacent. 00:14:37.600 --> 00:14:45.500 So, angles 2 and ABC are not adjacent angles, even though they have a common side. 00:14:45.500 --> 00:14:55.600 So, that would be my counter-example; the counter-example says that this is false, because this angle right here 00:14:55.600 --> 00:15:02.500 and this angle right here have a common side of BC, but they are not adjacent. 00:15:02.500 --> 00:15:10.700 So, keep that in mind--that it could be false--and then give a counter-example. 00:15:10.700 --> 00:15:20.100 The next one: An angle that measures 120 degrees is an obtuse angle. 00:15:20.100 --> 00:15:27.200 Let's write that as a conditional: An angle that measures 120 degrees is an obtuse angle, 00:15:27.200 --> 00:15:50.800 so if an angle measures 120 degrees, then it is an obtuse angle. 00:15:50.800 --> 00:16:06.200 Now, we know that that is true; if an angle measures 120 degrees, maybe like that right there (this is 120 degrees), then it is an obtuse angle. 00:16:06.200 --> 00:16:34.900 Let's write the converse now: If an angle is an obtuse angle, then it measures 120 degrees. 00:16:34.900 --> 00:16:38.700 We know that this is true; is the converse true? 00:16:38.700 --> 00:16:44.300 If an angle is an obtuse angle, does it measure 120 degrees? 00:16:44.300 --> 00:16:51.800 Well, can I draw another obtuse angle that is not 120 degrees--maybe a little bit bigger? 00:16:51.800 --> 00:16:55.900 This could be 130 degrees; that is still an obtuse angle. 00:16:55.900 --> 00:17:08.900 So then, this right here would be my counter-example, because this is false, and I am showing an example of when the statement is not true. 00:17:08.900 --> 00:17:19.900 The next one: Two angles with the same measures are congruent. 00:17:19.900 --> 00:17:44.100 So, if two angles have the same measure, then they are congruent. 00:17:44.100 --> 00:18:22.600 The converse (and this just means "congruent"): If two angles are congruent, then they have the same measure. 00:18:22.600 --> 00:18:32.800 If two angles have the same measure...there is an angle, and here is another angle...they are the same. 00:18:32.800 --> 00:18:40.700 They have the same measure, meaning that...let's say this is 40 degrees; this is 40 degrees. 00:18:40.700 --> 00:18:57.700 Then, they are congruent; so if this is ABC, and this is DEF, I know that, since the measure of angle ABC 00:18:57.700 --> 00:19:09.100 is 40, and the measure of angle DEF is 40, they have the same measure; then they are congruent. 00:19:09.100 --> 00:19:23.100 So then, angle ABC is congruent to angle DEF; this is true. 00:19:23.100 --> 00:19:32.700 If two angles are congruent, then they have the same measure; that is true, also. 00:19:32.700 --> 00:19:43.100 That means that the measure of angle ABC equals the measure of angle DEF. 00:19:43.100 --> 00:19:59.400 And this is the definition of congruency; so you can go from congruent angles to having the same measure; so this is also true. 00:19:59.400 --> 00:20:06.900 The next one, the second statement, is the inverse statement. 00:20:06.900 --> 00:20:18.200 This one uses what is called negation; now, when you negate something, you are saying that it is not that. 00:20:18.200 --> 00:20:31.100 So, if you have p, the hypothesis, then you can say not p; and it is represented by this little symbol right here: this means "not p." 00:20:31.100 --> 00:20:41.600 So, if a given statement is "an angle is obtuse," then the negated statement would be "an angle is not obtuse." 00:20:41.600 --> 00:20:51.300 That is all you are doing; and what inverse statements do is negate both the hypothesis and the conclusion. 00:20:51.300 --> 00:21:09.100 So, you are saying, "if not p, then not q"; your conditional was "if p, then q"; the inverse statement is going to be "if not p, then not q." 00:21:09.100 --> 00:21:17.800 And that is how you are going to write it: like this: not p to not q; and this is the inverse. 00:21:17.800 --> 00:21:20.800 Here is a given statement, "If it is raining, then I use an umbrella." 00:21:20.800 --> 00:21:26.000 The inverse is going to be, "If it is not raining, then I do not use an umbrella." 00:21:26.000 --> 00:21:32.700 Remember: for converse, all you do is interchange the hypothesis and conclusion. 00:21:32.700 --> 00:21:39.100 With the inverse, you don't interchange anything; all that you are going to do is negate both statements, the hypothesis and the conclusion. 00:21:39.100 --> 00:21:47.800 If it is not raining, then I do not use an umbrella. 00:21:47.800 --> 00:21:53.900 Write the inverse of each conditional; determine if it is true or false; if false, then give a counter-example. 00:21:53.900 --> 00:22:03.300 If three points lie on a line, then they are collinear. 00:22:03.300 --> 00:22:40.900 The inverse is going to be, "If three points do not lie on a line, then they are not collinear." 00:22:40.900 --> 00:22:50.300 We know that this right here, "If three points lie on a line, then they are collinear," is true; that is a true statement. 00:22:50.300 --> 00:23:00.600 If we have three points on a line, then they are going to be collinear. 00:23:00.600 --> 00:23:15.200 If three points do not lie on a line, then they are not collinear...let's see. 00:23:15.200 --> 00:23:24.700 If I have a line, and let's say one point is here; one point is here; and one is right here; three points do not lie on a line. 00:23:24.700 --> 00:23:29.600 Then, they are not collinear--is that true? That is true. 00:23:29.600 --> 00:23:35.100 If they don't lie on a line, then they are not collinear. 00:23:35.100 --> 00:23:41.500 The next one: Vertical angles are congruent. 00:23:41.500 --> 00:24:06.700 If you want to rewrite this as a conditional, you can: If angles are vertical, then they are congruent. 00:24:06.700 --> 00:24:34.600 The inverse statement: If angles are not vertical, then they are not congruent. 00:24:34.600 --> 00:24:44.200 If angles are vertical, then they are congruent; vertical angles would be this angle and this angle right here. 00:24:44.200 --> 00:24:48.600 So, they are vertical angles, and we know that they are congruent. 00:24:48.600 --> 00:24:54.500 Now, if angles are not vertical, then they are not congruent. 00:24:54.500 --> 00:25:04.300 Well, what if I have these angles right here? 00:25:04.300 --> 00:25:11.100 They are not vertical, but they can still be congruent, if this is 90 and this is 90. 00:25:11.100 --> 00:25:21.700 They have the same measure, so that means that they are congruent; so this would be false, and here is my counter-example. 00:25:21.700 --> 00:25:32.100 Now, the third statement is the contrapositive; and that is formed by doing both the converse and the inverse to it. 00:25:32.100 --> 00:25:38.800 You are going to exchange the hypothesis and the conclusion and negate both. 00:25:38.800 --> 00:25:45.800 Remember: the converse was where you exchange the hypothesis and the conclusion; in the inverse, you negate both the hypothesis and the conclusion. 00:25:45.800 --> 00:25:53.700 For a contrapositive, you are going to do both. 00:25:53.700 --> 00:26:02.900 If p, then q, was just the given conditional statement; but you are going to do "if not q, then not p." 00:26:02.900 --> 00:26:13.200 So, right here, we see that p and q have been interchanged; that is what you do for the converse. 00:26:13.200 --> 00:26:24.800 And then, not q and not p--that is negating both: so not q to not p is the contrapositive. 00:26:24.800 --> 00:26:28.200 The given statement: "If it is raining, then I use an umbrella." 00:26:28.200 --> 00:26:35.500 The contrapositive: "If I do not use an umbrella, then it is not raining." 00:26:35.500 --> 00:27:00.600 Here is my p; here is my q; if I do not use an umbrella...not: that means that I did negate; and this is the q statement; then it is not raining: negate p. 00:27:00.600 --> 00:27:08.200 Find the contrapositive of the conditional, and determine if it is true or false: Vertical angles are congruent. 00:27:08.200 --> 00:27:30.100 As a conditional, it is, "If angles are vertical, then they are congruent." 00:27:30.100 --> 00:28:10.100 The contrapositive is, "If"...then I need my q, my conclusion, negated, so, "If angles are not congruent, then they are not vertical." 00:28:10.100 --> 00:28:19.800 "If angles are vertical, then they are congruent" becomes "If angles are not congruent, then they are not vertical." 00:28:19.800 --> 00:28:28.700 So, this is a true statement; and then, if angles are not congruent, then they can't be vertical. 00:28:28.700 --> 00:28:44.200 This is also a true statement; now, for the contrapositive, when you have a conditional that is true, then the contrapositive will also be true. 00:28:44.200 --> 00:28:49.000 For the converse and the inverse, it could be true or false; but the contrapositive, 00:28:49.000 --> 00:28:56.200 as long as the original conditional is true, will always be true. 00:28:56.200 --> 00:29:06.900 If the given conditional statement is false, then the contrapositive will be false. 00:29:06.900 --> 00:29:22.000 The summary for this lesson: We have conditional statements; and this was one. 00:29:22.000 --> 00:29:47.700 You write statements in if/then form, and then, from here, you can write the converse, the inverse, and the contrapositive. 00:29:47.700 --> 00:30:07.100 If we know that "if p"--this is the hypothesis; the "then" statement is the conclusion; then the conditional statement is going to be, "if p, then q." 00:30:07.100 --> 00:30:15.000 Or I can also write it as p → q. 00:30:15.000 --> 00:30:22.300 The converse is when you switch the hypothesis and the conclusion. 00:30:22.300 --> 00:30:33.200 You are going to interchange them; so it is going to be if q, then p. 00:30:33.200 --> 00:30:44.800 And this is the converse: or you can write q → p; see how all they did was just switch. 00:30:44.800 --> 00:30:52.900 The inverse is when you negate; you are going to use negation. 00:30:52.900 --> 00:31:06.500 And this is if...back to p...back to the hypothesis...if not p, then not q. 00:31:06.500 --> 00:31:19.000 And that would be like that: not p, not q; it is important to know all of these, how to write it like this and like this. 00:31:19.000 --> 00:31:46.900 The contrapositive uses both: converse and inverse; and this would be if not q, then not p, not (wrong color) q, not p. 00:31:46.900 --> 00:32:03.500 And another thing to keep in mind: if this is a true statement, then remember that the contrapositive is always going to be a true statement. 00:32:03.500 --> 00:32:07.700 And then, for the converse and the inverse, it could be true, or it could be false. 00:32:07.700 --> 00:32:12.200 This could be true or false, and this could be true or false. 00:32:12.200 --> 00:32:21.100 But keep in mind that the contrapositive will be true, as long as the conditional statement is true. 00:32:21.100 --> 00:32:29.200 Let's do a few examples: Identify the hypothesis and the conclusion of each conditional statement. 00:32:29.200 --> 00:32:45.500 If it is sunny, then I will go to the beach: the hypothesis, again, follows the "if"; so "if it is sunny"--that is the condition--"if it is sunny." 00:32:45.500 --> 00:32:52.700 The conclusion is "then"--what is going to happen as a result: "I will go to the beach." 00:32:52.700 --> 00:32:59.800 The hypothesis is "it is sunny"; the conclusion is "I will go to the beach." 00:32:59.800 --> 00:33:10.100 The next one: If 3x - 5 = -11, then x = -2. 00:33:10.100 --> 00:33:25.000 My hypothesis is "if" this is the equation; "then" my solution is -2, and that is my conclusion. 00:33:25.000 --> 00:33:33.800 Write in if/then form: A piranha eats other fish. 00:33:33.800 --> 00:34:00.300 If the fish is a piranha, then it eats other fish. 00:34:00.300 --> 00:34:33.200 The next one: Equiangular triangles are equilateral: so if triangles are equiangular, then they are equilateral. 00:34:33.200 --> 00:34:40.000 And it is very important not to confuse the hypothesis and conclusion. 00:34:40.000 --> 00:34:52.600 A keyword here is "are," or "is"; something is something else--that is a good indicator of what the hypothesis is and what the conclusion is. 00:34:52.600 --> 00:35:01.000 OK, write the converse, inverse, and contrapositive, and determine if each is true or false. 00:35:01.000 --> 00:35:05.500 Then, if it is false, then give a counter-example. 00:35:05.500 --> 00:35:11.500 If you are 13 years old, then you are a teenager. 00:35:11.500 --> 00:35:41.100 The converse is when, remember, you interchange: If you are a teenager, then you are 13 years old. 00:35:41.100 --> 00:35:51.200 Is this true or false? Well, the given conditional, "If you are 13 years old, then you are a teenager," is true. 00:35:51.200 --> 00:35:55.900 How about this one, "If you are a teenager, then you are 13 years old"? 00:35:55.900 --> 00:35:59.200 Well, can you be 14 and still be a teenager? 00:35:59.200 --> 00:36:11.900 My counter-example will be showing that, if you are a teenager, then you can also be 14 years old; you can be 15; and so on, 00:36:11.900 --> 00:36:18.100 and still be considered a teenager; so this one is false. 00:36:18.100 --> 00:36:49.900 The inverse is when you negate both the hypothesis and the conclusion: "If you are not 13 years old, then you are not a teenager." 00:36:49.900 --> 00:37:02.000 Well, you can still be 12, and still be a teenager; so this one would be false. 00:37:02.000 --> 00:37:12.300 And then, the contrapositive is when you interchange them, and you negate both. 00:37:12.300 --> 00:37:34.600 If you are not a teenager, then you are not 13 years old. 00:37:34.600 --> 00:37:39.100 The contrapositive is, "If you are not a teenager, then you are not 13 years old." 00:37:39.100 --> 00:37:49.100 Well, if you are 13, you are considered a teenager; so if you are not a teenager, then you are not 13 years old; so this one is true. 00:37:49.100 --> 00:37:59.800 And again, since this is true, then the contrapositive is going to be true. 00:37:59.800 --> 00:38:06.000 Write the converse, inverse, and contrapositive, and determine if each is true or false. 00:38:06.000 --> 00:38:09.800 If it is false, then we are going to give a counter-example. 00:38:09.800 --> 00:38:19.800 Acute angles have measures less than 90 degrees. 00:38:19.800 --> 00:38:27.600 Let me change this to a conditional statement; or, as long as you know what the conditional statement is, 00:38:27.600 --> 00:38:32.300 then you can just go ahead and start writing the converse, inverse, and contrapositive. 00:38:32.300 --> 00:39:06.600 Acute angles have measures less than 90 degrees; a conditional statement is "if angles are acute, then they measure less than 90 degrees." 00:39:06.600 --> 00:39:39.200 And my converse is, "If angles measure 90 degrees, then they are acute." 00:39:39.200 --> 00:40:06.500 Let me just write them all out: the inverse is, "If angles are not acute, then they do not measure less than 90 degrees." 00:40:06.500 --> 00:40:12.800 Again, remember: inverse is when you just negate the hypothesis and the conclusion. 00:40:12.800 --> 00:40:23.100 You are going to make both of them the opposites; so if angles are acute, then the inverse would be "if angles are not acute." 00:40:23.100 --> 00:40:30.100 OK, and the contrapositive is when you are going to do both. 00:40:30.100 --> 00:40:36.800 You are going to interchange them, and you are going to negate both. 00:40:36.800 --> 00:41:06.200 "If angles do not measure less than 90 degrees, then they are not acute." 00:41:06.200 --> 00:41:18.000 So, this right here, I know, is a true statement: if angles are acute, then they measure less than 90. 00:41:18.000 --> 00:41:23.900 So, an acute angle is anything that is less than 90. 00:41:23.900 --> 00:41:34.000 And then, if angles measure...you know, I made a mistake here: if angles measure 90... 00:41:34.000 --> 00:41:43.700 sorry, it is not "measure 90," but "measure less than 90"...then they are acute. 00:41:43.700 --> 00:41:52.500 Is that true--"If angles measure less than 90, then they are acute"? Yes, that is true. 00:41:52.500 --> 00:41:59.000 The inverse is, "If angles are not acute, then they do not measure less than 90." 00:41:59.000 --> 00:42:05.600 That is true; if it is not acute, then it is either a right angle or an obtuse angle. 00:42:05.600 --> 00:42:12.800 If it is a right angle, then it measures exactly 90; and if it is an obtuse angle, then it has to measure more than 90. 00:42:12.800 --> 00:42:18.200 If it is not acute, then it is not going to measure less than 90; that is true. 00:42:18.200 --> 00:42:24.200 And the contrapositive is, "If angles do not measure less than 90, then they are not acute." 00:42:24.200 --> 00:42:35.400 So again, if they don't measure less than 90, then they can't be acute; then it is either going to be a right angle or an obtuse angle; so this is also true. 00:42:35.400 --> 00:42:41.700 And remember that, if the statement is true, then the contrapositive is also going to be true. 00:42:41.700 --> 00:42:45.800 That is it for this lesson; we will see you next time. 00:42:45.800 --> 00:42:47.000 Thank you for watching Educator.com!