WEBVTT mathematics/geometry/pyo
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Welcome back to Educator.com.
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This next lesson is on conditional statements.
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If/then statements are called **conditional statements**, or conditionals.
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When you have a statement in the form of if something, then something else, then that is considered a conditional statement.
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If you have a statement "I use an umbrella when it rains," you can rewrite it as a conditional in if/then form.
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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."
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When do you use an umbrella? When it rains, right? So, "If it is raining, then I use an umbrella."
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And that would be considered a conditional statement.
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If...this part right here, "If it is raining"--the phrase after the "if" is called the **hypothesis**.
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And then, the statement after the "then" is called the **conclusion**.
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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.
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That is what is going to result from the hypothesis.
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You can also think of the hypothesis as p; p is the hypothesis, and q is the conclusion.
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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.
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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."
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Again, the statement after the "if" is the hypothesis; the statement after the "then" is the conclusion.
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And then, it is if p, then q; you can also denote it as this, p → q; and that is "p implies q."
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Now, you could write this in a couple of different ways; you don't always have to write it "if" and "then."
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And it is still going to be considered a conditional: back to this example, "If it is raining, then I use an umbrella."
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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.
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"If it is raining, then I use an umbrella" can also be "If it is raining, I will just use an umbrella."
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You can also write it using "when" instead of the "if"; you are going to use the word when instead of if.
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"When it is raining, then I use an umbrella": just because you don't see an if there...
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this is still going to be the hypothesis, and then this is the conclusion.
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You can also reword it by stating the hypothesis at the end of it: "I use an umbrella if it is raining."
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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.
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Just keep that in mind: the hypothesis doesn't always have to be in the front.
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Let's identify the hypothesis and the conclusion: the first one: I am going to make the hypothesis red, and the conclusion will be...
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If it is Tuesday, then Phil plays tennis: well, the hypothesis, I know, is "if it is Tuesday."
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So, "it is Tuesday" will be the hypothesis; then what is going to happen as a result?
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Phil is going to play tennis; that is the conclusion.
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If it is Tuesday, then Phil plays tennis.
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The next one: Three points that lie on a line are collinear.
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Now, this is not written as a conditional statement; so let's rewrite this in if/then form.
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Three points that lie on a line are collinear; If three points lie on a line, then they are collinear.
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My hypothesis, then, is "three points lie on a line"; and then, my conclusion is going to be "then they are collinear."
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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.
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The next one: You are at least 21 years old if you are an adult.
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If you look at this, I see an "if" right here; so "you are at least 21 years old," *if* "you are an adult."
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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.
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If you are an adult, then you are at least 21 years old.
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These examples, we are going to write in if/then form; adjacent angles have a common vertex.
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If angles are adjacent, then they have a common vertex.
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The next one: Glass objects are fragile; what is fragile?--glass objects.
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So, if the objects...you can write this a couple of different ways.
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You can say, "If the objects are made of glass"; you can say, "If these objects are glass objects..."
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I am just going to say, "If the objects are glass, then..." what?..."they are fragile."
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And the third one: An angle is obtuse if its measure is greater than 90 degrees.
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If..."its"...we want to rewrite this word; if an angle measures greater than 90 degrees, then it is obtuse.
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OK, when we are given a conditional, we can write those given statements in three other forms,
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meaning that we can change the conditionals around in three different ways.
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And the first way is the converse way: converse statements.
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Oh, and then, we are going to go over each of these separately; so converse statements is the first one,
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then inverse statements, then contrapositive statements; so just keep in mind that there are three different ways.
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And the first one, converse statements, is when you interchange the hypothesis and the conclusion.
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So, remember how we had if p, then q; the hypothesis is p; the conclusion is q.
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When you switch the p and the q, that is a converse; so what happens then is: it becomes if q, then p.
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The if and then are still the same; you are still writing the conditional; but you are just switching the hypothesis and the conclusion.
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And when you write the converse, it doesn't necessarily have to be true.
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It can be true or false; so again, this is going to be if q, then p.
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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.
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Here is an example: If it is raining, then I use an umbrella--that is the given conditional statement.
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Then, the converse, by switching: this is the hypothesis; "then I use an umbrella"--that is the conclusion.
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You are going to interchange these two; so then, "If I use an umbrella, then it is raining."
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This is the converse statement, because you switched the hypothesis and the conclusion.
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This is p; this is q; so then, this became q, and this is p; the converse just interchanges them.
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Now, remember from the last section: we went over counter-examples.
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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.
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And that is when you can prove that it is false.
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And like I said earlier, converse statements are not necessarily true; they are going to be true or false.
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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.
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Write the converse of each given statement; decide if it is true or false; if false, write a counter-example.
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This one: Adjacent angles have a common side.
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Now, that is the given statement; we need to find the converse statement.
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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."
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Then, the converse is going to be, "If"...now remember: again, you are not putting "then" first;
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you are keeping the "if" and the "then" statement, but you are just interchanging these two;
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so, "if angles have a common side, then they are adjacent."
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Now, we know that this statement right here is true: "If angles are adjacent, then they have a common side"; that is true.
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"If angles have a common side, then they are adjacent": well, if I have an angle like this; this is A...angle ABC, D...
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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;
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that is the statement right here, and it is true.
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Now, if angles have a common side, then does that make them adjacent?
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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.
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This is their common side; but they are not adjacent.
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So, angles 2 and ABC are not adjacent angles, even though they have a common side.
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So, that would be my counter-example; the counter-example says that this is false, because this angle right here
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and this angle right here have a common side of BC, but they are not adjacent.
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So, keep that in mind--that it could be false--and then give a counter-example.
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The next one: An angle that measures 120 degrees is an obtuse angle.
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Let's write that as a conditional: An angle that measures 120 degrees is an obtuse angle,
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so if an angle measures 120 degrees, then it is an obtuse angle.
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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.
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Let's write the converse now: If an angle is an obtuse angle, then it measures 120 degrees.
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We know that this is true; is the converse true?
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If an angle is an obtuse angle, does it measure 120 degrees?
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Well, can I draw another obtuse angle that is not 120 degrees--maybe a little bit bigger?
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This could be 130 degrees; that is still an obtuse angle.
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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.
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The next one: Two angles with the same measures are congruent.
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So, if two angles have the same measure, then they are congruent.
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The converse (and this just means "congruent"): If two angles are congruent, then they have the same measure.
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If two angles have the same measure...there is an angle, and here is another angle...they are the same.
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They have the same measure, meaning that...let's say this is 40 degrees; this is 40 degrees.
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Then, they are congruent; so if this is ABC, and this is DEF, I know that, since the measure of angle ABC
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is 40, and the measure of angle DEF is 40, they have the same measure; then they are congruent.
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So then, angle ABC is congruent to angle DEF; this is true.
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If two angles are congruent, then they have the same measure; that is true, also.
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That means that the measure of angle ABC equals the measure of angle DEF.
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And this is the definition of congruency; so you can go from congruent angles to having the same measure; so this is also true.
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The next one, the second statement, is the **inverse statement**.
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This one uses what is called negation; now, when you negate something, you are saying that it is not that.
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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."
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So, if a given statement is "an angle is obtuse," then the negated statement would be "an angle is not obtuse."
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That is all you are doing; and what inverse statements do is negate both the hypothesis and the conclusion.
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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."
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And that is how you are going to write it: like this: not p to not q; and this is the inverse.
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Here is a given statement, "If it is raining, then I use an umbrella."
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The inverse is going to be, "If it is not raining, then I do not use an umbrella."
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Remember: for converse, all you do is interchange the hypothesis and conclusion.
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With the inverse, you don't interchange anything; all that you are going to do is negate both statements, the hypothesis and the conclusion.
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If it is not raining, then I do not use an umbrella.
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Write the inverse of each conditional; determine if it is true or false; if false, then give a counter-example.
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If three points lie on a line, then they are collinear.
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The inverse is going to be, "If three points do not lie on a line, then they are not collinear."
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We know that this right here, "If three points lie on a line, then they are collinear," is true; that is a true statement.
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If we have three points on a line, then they are going to be collinear.
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If three points do not lie on a line, then they are not collinear...let's see.
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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.
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Then, they are not collinear--is that true? That is true.
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If they don't lie on a line, then they are not collinear.
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The next one: Vertical angles are congruent.
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If you want to rewrite this as a conditional, you can: If angles are vertical, then they are congruent.
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The inverse statement: If angles are not vertical, then they are not congruent.
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If angles are vertical, then they are congruent; vertical angles would be this angle and this angle right here.
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So, they are vertical angles, and we know that they are congruent.
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Now, if angles are not vertical, then they are not congruent.
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Well, what if I have these angles right here?
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They are not vertical, but they can still be congruent, if this is 90 and this is 90.
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They have the same measure, so that means that they are congruent; so this would be false, and here is my counter-example.
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Now, the third statement is the **contrapositive**; and that is formed by doing both the converse and the inverse to it.
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You are going to exchange the hypothesis and the conclusion and negate both.
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Remember: the converse was where you exchange the hypothesis and the conclusion; in the inverse, you negate both the hypothesis and the conclusion.
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For a contrapositive, you are going to do both.
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If p, then q, was just the given conditional statement; but you are going to do "if not q, then not p."
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So, right here, we see that p and q have been interchanged; that is what you do for the converse.
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And then, not q and not p--that is negating both: so not q to not p is the contrapositive.
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The given statement: "If it is raining, then I use an umbrella."
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The contrapositive: "If I do not use an umbrella, then it is not raining."
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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.
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Find the contrapositive of the conditional, and determine if it is true or false: Vertical angles are congruent.
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As a conditional, it is, "If angles are vertical, then they are congruent."
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The contrapositive is, "If"...then I need my q, my conclusion, negated, so, "If angles are not congruent, then they are not vertical."
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"If angles are vertical, then they are congruent" becomes "If angles are not congruent, then they are not vertical."
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So, this is a true statement; and then, if angles are not congruent, then they can't be vertical.
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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.
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For the converse and the inverse, it could be true or false; but the contrapositive,
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as long as the original conditional is true, will always be true.
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If the given conditional statement is false, then the contrapositive will be false.
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The summary for this lesson: We have conditional statements; and this was one.
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You write statements in if/then form, and then, from here, you can write the converse, the inverse, and the contrapositive.
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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."
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Or I can also write it as p → q.
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The converse is when you switch the hypothesis and the conclusion.
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You are going to interchange them; so it is going to be if q, then p.
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And this is the converse: or you can write q → p; see how all they did was just switch.
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The inverse is when you negate; you are going to use negation.
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And this is if...back to p...back to the hypothesis...if not p, then not q.
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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.
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The contrapositive uses both: converse and inverse; and this would be if not q, then not p, not (wrong color) q, not p.
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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.
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And then, for the converse and the inverse, it could be true, or it could be false.
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This could be true or false, and this could be true or false.
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But keep in mind that the contrapositive will be true, as long as the conditional statement is true.
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Let's do a few examples: Identify the hypothesis and the conclusion of each conditional statement.
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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."
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The conclusion is "then"--what is going to happen as a result: "I will go to the beach."
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The hypothesis is "it is sunny"; the conclusion is "I will go to the beach."
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The next one: If 3x - 5 = -11, then x = -2.
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My hypothesis is "if" this is the equation; "then" my solution is -2, and that is my conclusion.
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Write in if/then form: A piranha eats other fish.
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If the fish is a piranha, then it eats other fish.
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The next one: Equiangular triangles are equilateral: so if triangles are equiangular, then they are equilateral.
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And it is very important not to confuse the hypothesis and conclusion.
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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.
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OK, write the converse, inverse, and contrapositive, and determine if each is true or false.
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Then, if it is false, then give a counter-example.
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If you are 13 years old, then you are a teenager.
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The converse is when, remember, you interchange: If you are a teenager, then you are 13 years old.
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Is this true or false? Well, the given conditional, "If you are 13 years old, then you are a teenager," is true.
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How about this one, "If you are a teenager, then you are 13 years old"?
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Well, can you be 14 and still be a teenager?
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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,
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and still be considered a teenager; so this one is false.
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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."
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Well, you can still be 12, and still be a teenager; so this one would be false.
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And then, the contrapositive is when you interchange them, and you negate both.
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If you are not a teenager, then you are not 13 years old.
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The contrapositive is, "If you are not a teenager, then you are not 13 years old."
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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.
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And again, since this is true, then the contrapositive is going to be true.
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Write the converse, inverse, and contrapositive, and determine if each is true or false.
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If it is false, then we are going to give a counter-example.
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Acute angles have measures less than 90 degrees.
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Let me change this to a conditional statement; or, as long as you know what the conditional statement is,
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then you can just go ahead and start writing the converse, inverse, and contrapositive.
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Acute angles have measures less than 90 degrees; a conditional statement is "if angles are acute, then they measure less than 90 degrees."
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And my converse is, "If angles measure 90 degrees, then they are acute."
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Let me just write them all out: the inverse is, "If angles are not acute, then they do not measure less than 90 degrees."
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Again, remember: inverse is when you just negate the hypothesis and the conclusion.
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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."
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OK, and the contrapositive is when you are going to do both.
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You are going to interchange them, and you are going to negate both.
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"If angles do not measure less than 90 degrees, then they are not acute."
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So, this right here, I know, is a true statement: if angles are acute, then they measure less than 90.
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So, an acute angle is anything that is less than 90.
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And then, if angles measure...you know, I made a mistake here: if angles measure 90...
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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.
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The inverse is, "If angles are not acute, then they do not measure less than 90."
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That is true; if it is not acute, then it is either a right angle or an obtuse angle.
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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.
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If it is not acute, then it is not going to measure less than 90; that is true.
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And the contrapositive is, "If angles do not measure less than 90, then they are not acute."
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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.
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And remember that, if the statement is true, then the contrapositive is also going to be true.
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That is it for this lesson; we will see you next time.
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