Geometry Conditional Statements

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This next lesson is on conditional statements.0002

If/then statements are called conditional statements, or conditionals.0006

When you have a statement in the form of if something, then something else, then that is considered a conditional statement.0014

If you have a statement "I use an umbrella when it rains," you can rewrite it as a conditional in if/then form.0025

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."0034

When do you use an umbrella? When it rains, right? So, "If it is raining, then I use an umbrella."0044

And that would be considered a conditional statement.0051

If...this part right here, "If it is raining"--the phrase after the "if" is called the hypothesis.0057

And then, the statement after the "then" is called the conclusion.0072

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.0080

That is what is going to result from the hypothesis.0087

You can also think of the hypothesis as p; p is the hypothesis, and q is the conclusion.0091

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.0105

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."0118

Again, the statement after the "if" is the hypothesis; the statement after the "then" is the conclusion.0134

And then, it is if p, then q; you can also denote it as this, p → q; and that is "p implies q."0140

Now, you could write this in a couple of different ways; you don't always have to write it "if" and "then."0151

And it is still going to be considered a conditional: back to this example, "If it is raining, then I use an umbrella."0158

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.0165

"If it is raining, then I use an umbrella" can also be "If it is raining, I will just use an umbrella."0177

You can also write it using "when" instead of the "if"; you are going to use the word when instead of if.0184

"When it is raining, then I use an umbrella": just because you don't see an if there...0192

this is still going to be the hypothesis, and then this is the conclusion.0199

You can also reword it by stating the hypothesis at the end of it: "I use an umbrella if it is raining."0205

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.0213

Just keep that in mind: the hypothesis doesn't always have to be in the front.0222

Let's identify the hypothesis and the conclusion: the first one: I am going to make the hypothesis red, and the conclusion will be...0234

If it is Tuesday, then Phil plays tennis: well, the hypothesis, I know, is "if it is Tuesday."0247

So, "it is Tuesday" will be the hypothesis; then what is going to happen as a result?0257

Phil is going to play tennis; that is the conclusion.0264

If it is Tuesday, then Phil plays tennis.0268

The next one: Three points that lie on a line are collinear.0271

Now, this is not written as a conditional statement; so let's rewrite this in if/then form.0280

Three points that lie on a line are collinear; If three points lie on a line, then they are collinear.0290

My hypothesis, then, is "three points lie on a line"; and then, my conclusion is going to be "then they are collinear."0319

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.0331

The next one: You are at least 21 years old if you are an adult.0339

If you look at this, I see an "if" right here; so "you are at least 21 years old," if "you are an adult."0350

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.0359

If you are an adult, then you are at least 21 years old.0368

These examples, we are going to write in if/then form; adjacent angles have a common vertex.0379

If angles are adjacent, then they have a common vertex.0393

The next one: Glass objects are fragile; what is fragile?--glass objects.0418

So, if the objects...you can write this a couple of different ways.0424

You can say, "If the objects are made of glass"; you can say, "If these objects are glass objects..."0433

I am just going to say, "If the objects are glass, then..." what?..."they are fragile."0443

And the third one: An angle is obtuse if its measure is greater than 90 degrees.0459

If..."its"...we want to rewrite this word; if an angle measures greater than 90 degrees, then it is obtuse.0474

OK, when we are given a conditional, we can write those given statements in three other forms,0522

meaning that we can change the conditionals around in three different ways.0537

And the first way is the converse way: converse statements.0543

Oh, and then, we are going to go over each of these separately; so converse statements is the first one,0550

then inverse statements, then contrapositive statements; so just keep in mind that there are three different ways.0554

And the first one, converse statements, is when you interchange the hypothesis and the conclusion.0559

So, remember how we had if p, then q; the hypothesis is p; the conclusion is q.0566

When you switch the p and the q, that is a converse; so what happens then is: it becomes if q, then p.0571

The if and then are still the same; you are still writing the conditional; but you are just switching the hypothesis and the conclusion.0583

And when you write the converse, it doesn't necessarily have to be true.0593

It can be true or false; so again, this is going to be if q, then p.0598

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.0605

Here is an example: If it is raining, then I use an umbrella--that is the given conditional statement.0618

Then, the converse, by switching: this is the hypothesis; "then I use an umbrella"--that is the conclusion.0625

You are going to interchange these two; so then, "If I use an umbrella, then it is raining."0632

This is the converse statement, because you switched the hypothesis and the conclusion.0643

This is p; this is q; so then, this became q, and this is p; the converse just interchanges them.0650

Now, remember from the last section: we went over counter-examples.0665

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.0670

And that is when you can prove that it is false.0683

And like I said earlier, converse statements are not necessarily true; they are going to be true or false.0689

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.0694

Write the converse of each given statement; decide if it is true or false; if false, write a counter-example.0704

This one: Adjacent angles have a common side.0712

Now, that is the given statement; we need to find the converse statement.0718

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."0722

Then, the converse is going to be, "If"...now remember: again, you are not putting "then" first;0754

you are keeping the "if" and the "then" statement, but you are just interchanging these two;0767

so, "if angles have a common side, then they are adjacent."0773

Now, we know that this statement right here is true: "If angles are adjacent, then they have a common side"; that is true.0797

"If angles have a common side, then they are adjacent": well, if I have an angle like this; this is A...angle ABC, D...0805

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;0824

that is the statement right here, and it is true.0842

Now, if angles have a common side, then does that make them adjacent?0847

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.0851

This is their common side; but they are not adjacent.0872

So, angles 2 and ABC are not adjacent angles, even though they have a common side.0877

So, that would be my counter-example; the counter-example says that this is false, because this angle right here0885

and this angle right here have a common side of BC, but they are not adjacent.0895

So, keep that in mind--that it could be false--and then give a counter-example.0902

The next one: An angle that measures 120 degrees is an obtuse angle.0911

Let's write that as a conditional: An angle that measures 120 degrees is an obtuse angle,0920

so if an angle measures 120 degrees, then it is an obtuse angle.0927

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.0951

Let's write the converse now: If an angle is an obtuse angle, then it measures 120 degrees.0966

We know that this is true; is the converse true?0995

If an angle is an obtuse angle, does it measure 120 degrees?0999

Well, can I draw another obtuse angle that is not 120 degrees--maybe a little bit bigger?1004

This could be 130 degrees; that is still an obtuse angle.1012

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.1016

The next one: Two angles with the same measures are congruent.1029

So, if two angles have the same measure, then they are congruent.1040

The converse (and this just means "congruent"): If two angles are congruent, then they have the same measure.1064

If two angles have the same measure...there is an angle, and here is another angle...they are the same.1102

They have the same measure, meaning that...let's say this is 40 degrees; this is 40 degrees.1113

Then, they are congruent; so if this is ABC, and this is DEF, I know that, since the measure of angle ABC1121

is 40, and the measure of angle DEF is 40, they have the same measure; then they are congruent.1138

So then, angle ABC is congruent to angle DEF; this is true.1149

If two angles are congruent, then they have the same measure; that is true, also.1163

That means that the measure of angle ABC equals the measure of angle DEF.1173

And this is the definition of congruency; so you can go from congruent angles to having the same measure; so this is also true.1183

The next one, the second statement, is the inverse statement.1199

This one uses what is called negation; now, when you negate something, you are saying that it is not that.1207

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."1218

So, if a given statement is "an angle is obtuse," then the negated statement would be "an angle is not obtuse."1231

That is all you are doing; and what inverse statements do is negate both the hypothesis and the conclusion.1241

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."1251

And that is how you are going to write it: like this: not p to not q; and this is the inverse.1269

Here is a given statement, "If it is raining, then I use an umbrella."1278

The inverse is going to be, "If it is not raining, then I do not use an umbrella."1281

Remember: for converse, all you do is interchange the hypothesis and conclusion.1286

With the inverse, you don't interchange anything; all that you are going to do is negate both statements, the hypothesis and the conclusion.1293

If it is not raining, then I do not use an umbrella.1299

Write the inverse of each conditional; determine if it is true or false; if false, then give a counter-example.1308

If three points lie on a line, then they are collinear.1314

The inverse is going to be, "If three points do not lie on a line, then they are not collinear."1323

We know that this right here, "If three points lie on a line, then they are collinear," is true; that is a true statement.1361

If we have three points on a line, then they are going to be collinear.1370

If three points do not lie on a line, then they are not collinear...let's see.1380

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.1395

Then, they are not collinear--is that true? That is true.1405

If they don't lie on a line, then they are not collinear.1409

The next one: Vertical angles are congruent.1415

If you want to rewrite this as a conditional, you can: If angles are vertical, then they are congruent.1421

The inverse statement: If angles are not vertical, then they are not congruent.1447

If angles are vertical, then they are congruent; vertical angles would be this angle and this angle right here.1474

So, they are vertical angles, and we know that they are congruent.1484

Now, if angles are not vertical, then they are not congruent.1488

Well, what if I have these angles right here?1494

They are not vertical, but they can still be congruent, if this is 90 and this is 90.1504

They have the same measure, so that means that they are congruent; so this would be false, and here is my counter-example.1511

Now, the third statement is the contrapositive; and that is formed by doing both the converse and the inverse to it.1522

You are going to exchange the hypothesis and the conclusion and negate both.1532

Remember: the converse was where you exchange the hypothesis and the conclusion; in the inverse, you negate both the hypothesis and the conclusion.1539

For a contrapositive, you are going to do both.1546

If p, then q, was just the given conditional statement; but you are going to do "if not q, then not p."1554

So, right here, we see that p and q have been interchanged; that is what you do for the converse.1563

And then, not q and not p--that is negating both: so not q to not p is the contrapositive.1573

The given statement: "If it is raining, then I use an umbrella."1585

The contrapositive: "If I do not use an umbrella, then it is not raining."1588

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.1595

Find the contrapositive of the conditional, and determine if it is true or false: Vertical angles are congruent.1620

As a conditional, it is, "If angles are vertical, then they are congruent."1628

The contrapositive is, "If"...then I need my q, my conclusion, negated, so, "If angles are not congruent, then they are not vertical."1650

"If angles are vertical, then they are congruent" becomes "If angles are not congruent, then they are not vertical."1690

So, this is a true statement; and then, if angles are not congruent, then they can't be vertical.1700

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.1709

For the converse and the inverse, it could be true or false; but the contrapositive,1724

as long as the original conditional is true, will always be true.1729

If the given conditional statement is false, then the contrapositive will be false.1736

The summary for this lesson: We have conditional statements; and this was one.1747

You write statements in if/then form, and then, from here, you can write the converse, the inverse, and the contrapositive.1762

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."1788

Or I can also write it as p → q.1807

The converse is when you switch the hypothesis and the conclusion.1815

You are going to interchange them; so it is going to be if q, then p.1822

And this is the converse: or you can write q → p; see how all they did was just switch.1833

The inverse is when you negate; you are going to use negation.1845

And this is if...back to p...back to the hypothesis...if not p, then not q.1853

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.1866

The contrapositive uses both: converse and inverse; and this would be if not q, then not p, not (wrong color) q, not p.1879

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.1907

And then, for the converse and the inverse, it could be true, or it could be false.1923

This could be true or false, and this could be true or false.1928

But keep in mind that the contrapositive will be true, as long as the conditional statement is true.1932

Let's do a few examples: Identify the hypothesis and the conclusion of each conditional statement.1941

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."1949

The conclusion is "then"--what is going to happen as a result: "I will go to the beach."1965

The hypothesis is "it is sunny"; the conclusion is "I will go to the beach."1973

The next one: If 3x - 5 = -11, then x = -2.1980

My hypothesis is "if" this is the equation; "then" my solution is -2, and that is my conclusion.1990

Write in if/then form: A piranha eats other fish.2005

If the fish is a piranha, then it eats other fish.2014

The next one: Equiangular triangles are equilateral: so if triangles are equiangular, then they are equilateral.2040

And it is very important not to confuse the hypothesis and conclusion.2073

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.2080

OK, write the converse, inverse, and contrapositive, and determine if each is true or false.2092

Then, if it is false, then give a counter-example.2101

If you are 13 years old, then you are a teenager.2105

The converse is when, remember, you interchange: If you are a teenager, then you are 13 years old.2111

Is this true or false? Well, the given conditional, "If you are 13 years old, then you are a teenager," is true.2141

How about this one, "If you are a teenager, then you are 13 years old"?2151

Well, can you be 14 and still be a teenager?2156

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,2159

and still be considered a teenager; so this one is false.2172

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."2178

Well, you can still be 12, and still be a teenager; so this one would be false.2210

And then, the contrapositive is when you interchange them, and you negate both.2222

If you are not a teenager, then you are not 13 years old.2232

The contrapositive is, "If you are not a teenager, then you are not 13 years old."2254

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.2259

And again, since this is true, then the contrapositive is going to be true.2269

Write the converse, inverse, and contrapositive, and determine if each is true or false.2280

If it is false, then we are going to give a counter-example.2286

Acute angles have measures less than 90 degrees.2290

Let me change this to a conditional statement; or, as long as you know what the conditional statement is,2300

then you can just go ahead and start writing the converse, inverse, and contrapositive.2307

Acute angles have measures less than 90 degrees; a conditional statement is "if angles are acute, then they measure less than 90 degrees."2312

And my converse is, "If angles measure 90 degrees, then they are acute."2346

Let me just write them all out: the inverse is, "If angles are not acute, then they do not measure less than 90 degrees."2379

Again, remember: inverse is when you just negate the hypothesis and the conclusion.2406

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."2413

OK, and the contrapositive is when you are going to do both.2423

You are going to interchange them, and you are going to negate both.2430

"If angles do not measure less than 90 degrees, then they are not acute."2437

So, this right here, I know, is a true statement: if angles are acute, then they measure less than 90.2466

So, an acute angle is anything that is less than 90.2478

And then, if angles measure...you know, I made a mistake here: if angles measure 90...2484

sorry, it is not "measure 90," but "measure less than 90"...then they are acute.2494

Is that true--"If angles measure less than 90, then they are acute"? Yes, that is true.2504

The inverse is, "If angles are not acute, then they do not measure less than 90."2512

That is true; if it is not acute, then it is either a right angle or an obtuse angle.2519

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 90.2525

If it is not acute, then it is not going to measure less than 90; that is true.2533

And the contrapositive is, "If angles do not measure less than 90, then they are not acute."2538

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.2544

And remember that, if the statement is true, then the contrapositive is also going to be true.2555

That is it for this lesson; we will see you next time.2562

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