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### Conditional Statements

- If-then statements are called conditional statements or conditionals.
- The conditional statement: If
*p,*then*q*. - Given statements can be written as condition statements in 3 other forms: converse statements, inverse statements, and contrapositive statements
- The converse of a given conditional interchanges the hypothesis and the conclusion. This statement can be true or false. If
*q*, then*p*. - The denial of a statement is called a negation. Inverse statements are formed by negating both the hypothesis and conclusion.
- A contrapositive statement is formed by exchanging and negating the hypothesis and conclusion of the given conditional.

### Conditional Statements

If I eat all the food on the plate, then I will be sick.

the conclusion is : I will be sick.

Dogs are lovely.

- IF then form: If they are dogs, then they are lovely.

conclusion: they are lovely.

Linear angles share one side.

If two angles share a vertex and a side, then they are adjacent angles.

True

If the two angles are adjacent, then they share one side.

False.

Acute angles are less than 90

^{o}

^{}.

- The statement in if then form: If angles are acute, then they are less than 90
^{o}.

^{o}, then they are not acute angles.

True.

If p, then q.

Inverse: If not p, then not q. True or False.

Contropositive: If not q, then not p. True.

If 5x − 8 = 2, then x = 2.

True.

Right triangle has an angle measures 90.

- IF then form: If a triangle is a right triangle, then it has an angle measures 90.

True.

If line AB passes through point C, then point C is on line AB.

True.

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Conditional Statements

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro
- If Then Statements
- Other Forms
- Identify the Hypothesis and Conclusion
- Example 1: Hypothesis and Conclusion
- Example 2: Hypothesis and Conclusion
- Example 3: Hypothesis and Conclusion
- Write in If Then Form
- Other Statements
- Converse Statements
- Converses and Counterexamples
- Converses and Counterexamples
- Example 1: Converses and Counterexamples
- Example 2: Converses and Counterexamples
- Example 3: Converses and Counterexamples
- Inverse Statement
- Inverse Statement
- Contrapositive Statement
- Contrapositive Statement
- Summary
- Extra Example 1: Hypothesis and Conclusion
- Extra Example 2: If-Then Form
- Extra Example 3: Converse, Inverse, and Contrapositive
- Extra Example 4: Converse, Inverse, and Contrapositive

- Intro 0:00
- If Then Statements 0:05
- If Then Statements
- Other Forms 2:29
- Example: Without Then
- Example: Using When
- Example: Hypothesis
- Identify the Hypothesis and Conclusion 3:52
- Example 1: Hypothesis and Conclusion
- Example 2: Hypothesis and Conclusion
- Example 3: Hypothesis and Conclusion
- Write in If Then Form 6:16
- Example 1: Write in If Then Form
- Example 2: Write in If Then Form
- Example 3: Write in If Then Form
- Other Statements 8:40
- Other Statements
- Converse Statements 9:18
- Converse Statements
- Converses and Counterexamples 11:04
- Converses and Counterexamples
- Example 1: Converses and Counterexamples
- Example 2: Converses and Counterexamples
- Example 3: Converses and Counterexamples
- Inverse Statement 19:58
- Definition and Example
- Inverse Statement 21:46
- Example 1: Inverse and Counterexample
- Example 2: Inverse and Counterexample
- Contrapositive Statement 25:20
- Definition and Example
- Contrapositive Statement 26:58
- Example: Contrapositive Statement
- Summary 29:03
- Summary of Lesson
- Extra Example 1: Hypothesis and Conclusion 32:20
- Extra Example 2: If-Then Form 33:23
- Extra Example 3: Converse, Inverse, and Contrapositive 34:54
- Extra Example 4: Converse, Inverse, and Contrapositive 37:56

### Geometry Online Course

### Transcription: Conditional Statements

*Welcome back to Educator.com.*0000

*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 here*0885

*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 ABC*1121

*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

*Thank you for watching Educator.com!*2566

0 answers

Post by Khanh Nguyen on April 28, 2015

On question 10, it says ture, I think it should be changed to "true". Could you change it? Thank You

0 answers

Post by Shahram Ahmadi N. Emran on July 15, 2013

In Example 3 of this lecture/lesson:

The inverse statement is incorrect and is not false.

The correct inverse statement should be If you are not 13 years old, then you are not a teenager and is a TRUE statement. Please verify this answer.

1 answer

Last reply by: Professor Pyo

Fri Aug 2, 2013 1:22 AM

Post by Shahram Ahmadi N. Emran on July 15, 2013

In Example 3 of this lecture/lesson:

The inverse statement is incorrect and is not false.

The correct inverse statement should be If you are not 13 years old, then you are not a teenager and is a TRUE statement.

1 answer

Last reply by: Professor Pyo

Fri Aug 2, 2013 1:13 AM

Post by julius mogyorossy on July 14, 2013

When you talk about vertical angles you are talking about angles that are opposite of each other, correct. I don't see how you proved that = = congruent, I can't see your logic. It just seems that they are just words that are defined to be the same thing.

0 answers

Post by Kenneth Montfort on February 20, 2013

In the converse and counterexamples practice page (11:04) - can you use a linear pair definition as a counterexample for the first problem?

0 answers

Post by Charlie Jiang on September 11, 2012

good

0 answers

Post by Jorge Guerrero on April 22, 2012

Great lesson! Very useful for law in argumentative stages of a case. Thank you for the skill to apply.

0 answers

Post by JAMIE CHEN on August 1, 2011

very nice work done by mary. her bright and lively personality livens up such a repetitive lesson. C:

+1 comfirm