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### Deductive Reasoning

- Deductive reasoning: The process of reasoning logically from given statements to a conclusion. If given statements are true, then deductive reasoning produces a true conclusion.
- Inductive reasoning uses examples and patterns to make conjectures
- Deductive reasoning uses logic and rules to make a conclusion
- Law of Detachment: If a conditional is true and its hypothesis is true, then the conclusion is true. If p→q, is true and p is true, then q is true.
- Law of Syllogism: If p→q and q→r are true conditionals, then p→r is also true. This is similar to the Transitive Property of Equality.

### Deductive Reasoning

Make a conclusion for this condition: David is late for the exam.

1. If ∠1 and ∠2 are linear pairs, then m∠1 + m∠2 = 180

^{o}.

2. If m∠1 + m∠2 = 180

^{o}, then they are supplementary angles.

1. If a point is on line m, it is colinear with points A and B.

2. Point C is on line m.

1. If an angle is smaller than 90

^{o}, it is an acute angle.

2. ∠1 is 45

^{o}.

1. If you want to get A for this class, you have to be the top 10% of your class.

2. If you want to be the top 10% of your class, you have to work hard.

1. Two angles are complementary if they add up to 90

^{o}.

2. m∠1 + m∠2 = 90

^{o}.

3. ∠1 and ∠2 are complementary.

1. If you jump out from the window, you will break your leg.

2. If you break your leg, you can not go to school.

3. If you jump out from the window, you can not go to school.

1. If two lines intersect at a point, then they are coplanar.

2. If two lines are coplanar, all the points on them are coplanar.

3. If two lines intersect at a point, then all the points on them are coplanar.

1. If a line passes through a point, then the line contains the point.

2. Line m passes through point A.

3. Line m contains point A.

1. If a triangle has a right angle, then it is a right triangle.

2. A right triangle follows the Pythagorean theorem.

3. If a triangle has a right angle, then it follows the Pythagorean theorem.

*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

### Deductive Reasoning

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
- Deductive Reasoning
- Inductive vs. Deductive
- Law of Detachment
- Law of Syllogism
- Using Laws of Logic
- Using Laws of Logic, cont.
- Extra Example 1: Determine the Conclusion and Law
- Extra Example 2: Determine the Conclusion and Law
- Extra Example 3: Determine the Logic and Law
- Extra Example 4: Determine the Logic and Law

- Intro 0:00
- Deductive Reasoning 0:06
- Definition of Deductive Reasoning
- Inductive vs. Deductive 2:51
- Inductive Reasoning
- Deductive reasoning
- Law of Detachment 3:47
- Law of Detachment
- Examples of Law of Detachment
- Law of Syllogism 7:32
- Law of Syllogism
- Example 1: Making a Conclusion
- Example 2: Making a Conclusion
- Using Laws of Logic 14:12
- Example 1: Determine the Logic
- Example 2: Determine the Logic
- Using Laws of Logic, cont. 18:47
- Example 3: Determine the Logic
- Example 4: Determine the Logic
- Extra Example 1: Determine the Conclusion and Law 22:12
- Extra Example 2: Determine the Conclusion and Law 25:39
- Extra Example 3: Determine the Logic and Law 29:50
- Extra Example 4: Determine the Logic and Law 31:27

### Geometry Online Course

### Transcription: Deductive Reasoning

*Welcome back to Educator.com.*0000

*This next lesson is on deductive reasoning.*0002

* Deductive reasoning is the process of reasoning logically--that is the keyword right here, "logically."*0007

*You are going to use logic from given statements to form a conclusion.*0013

*If given statements are true, then deductive reasoning produces a true conclusion.*0019

*As long as we have statements that we can show as true, then based on those statements, we can come to a true conclusion.*0028

*And this is the process of deductive reasoning.*0044

*Many professions use deductive reasoning: doctors, when diagnosing a patient's illness...*0049

*A few lessons ago, we learned about inductive reasoning; that is the opposite of deductive reasoning.*0055

*Inductive reasoning uses, remember, examples and past experiences.*0061

*But for deductive reasoning, each situation is unique, and you are going to look at basically facts and truths--*0069

*anything that is true--to come up with that conclusion.*0078

*Doctors, when you diagnose a patient's illness, have to look at all the facts and what is there to be able to diagnose the illness correctly.*0082

*You don't want doctors to diagnose based on inductive reasoning, because then, as long as you have the same symptom, then you have the same illness.*0092

*For example, if you have a bruise, and you go in to see the doctor, inductive reasoning would suggest that,*0104

*well, since the last two patients that came in with bruises had some sort of illness,*0116

*you will have the same illness also, just because you have a bruise.*0124

*But deductive reasoning...again, you have to look at each unique situation, and looking at that individual,*0128

*and all of the given statements, all of what is true, the facts there--using that, the doctors will diagnose the patient's illness.*0137

*Carpenters, when deciding what materials are needed at a worksite: each time a carpenter has a different site, they need a different material.*0149

*So, deciding what materials to use at that specific worksite is considered deductive reasoning.*0160

*Again, inductive reasoning is using examples, past experiences, and patterns to make conjectures.*0174

*You make conjectures; you make guesses, using "Well, it happened this way the last five times,*0182

*so the sixth time, I can make a conjecture that it is going to happen again."*0190

*So, a conjecture is an educated guess.*0196

*Now, with deductive reasoning, you use logic, and you use rules, to come to a conclusion.*0199

*With inductive reasoning, you are just kind of guessing, just by patterns, what is going to come up next.*0208

*But with deductive reasoning, you are actually looking at the situation, and you are going to use logic;*0216

*and you are going to use rules and facts to make a conclusion, to base it on something.*0221

*The first law of logic is the Law of Detachment.*0230

*Now, if a conditional is true, and the hypothesis is true, then the conclusion is true.*0238

*If you look at this, it will be easier to understand.*0248

*This is the conditional statement: if p → q is true, and p is true, then q will be true.*0251

*As long as the conditional is true and the hypothesis is true, then the conclusion will be true.*0265

*Here is an example: If a student gets an A on the final exam, then the student will pass the course.*0273

*That is the conditional p to q: If a student...here is p; all of this is p, "a student gets an A on a final exam."*0280

*Then, the student will pass the course; here is q; so p to q is true.*0290

*Now, David gets an A on the geometry final; here, this is this p, so that is true,*0300

*because the conditional statement says that if a student gets an A on the final exam, then the student will pass the course.*0311

*Well, David got an A on the final exam; then what can you conclude--what kind of conclusion can you make?*0318

*It is that David, then, will pass the course.*0329

*So, this conditional was true; "If a student gets an A on a final exam, then the student will pass the course"--that is the given conditional.*0344

*Then, David gets an A on the final exam; that is part of this.*0355

*So, if he gets an A on the final exam, then you can say that he is going to pass the course, because that is what the conditional says, and the conditional is true.*0360

*The next example: If two numbers are odd, then their sum is even.*0370

*Two numbers are odd--here is p; their sum is even--here is q.*0380

*And then, 3 and 5 are odd numbers; this is based on p--this is all based on p.*0386

*p → q is true, and this right here, "3 and 5 are odd numbers"...then my conclusion is that the sum of 3 and 5 is even.*0394

*The sum is going to be even, then, because this is the conditional.*0420

*If two numbers are odd, then their sum is even; and 3 and 5 are odd numbers; then, the sum of 3 and 5 is even.*0424

*You are using the conditional and a hypothesis; then you are going to come to a conclusion.*0433

*And this is the Law of Detachment: if p → q is true, and p is true, then q is true.*0442

*The next one is the Law of Syllogism; this one is very similar to the transitive property of equality.*0454

*If you remember, from Algebra I, you learned the transitive property.*0461

*The transitive property says that, if A equals B, and B equals C, then A equals C.*0466

*If A equals B, and B equals C, then, since these two are equal, A equals C.*0475

*This is very similar to that: the Law of Syllogism says that if the conditional p → q is true,*0491

*and q → r, that conditional, is true, then p → r is also true.*0502

*So then, here you have two different conditional statements.*0511

*You have p → q, and then you have q → r; now remember, this q and this q have to be the same.*0515

*p → q is true; q → r is true; this is a different conclusion; then, this hypothesis, p, to this conclusion, r, is going to also be true, just like this one.*0522

*A to B and B to C...then A is equal to C.*0537

*Let's just do a couple of examples: Using the two given statements, make a conclusion, if possible.*0544

*If M is the midpoint of segment AB, then AM is equal to MB.*0554

*If I have segment AB, and M is the midpoint (this is M), then AM is equal to MB.*0563

*If the measures of two segments are equal, then they are congruent.*0580

*Here, this segment and this segment are equal; right here, that is what it says.*0602

*Here is AM, and here is MB, and they are equal to each other; then, they are congruent.*0609

*So, all of this right here--this is all p; this first one would be p → q; and then, this is q.*0616

*This next one, "The measures of two segments are equal," is saying the same thing as this right here: AM = MB/two segments are equal.*0636

*So then, this is using q; then they are congruent--now, this is a new conclusion, so this is r; so this is q → r.*0651

*So, my conclusion...see, right here, the Law of Syllogism says p → q; there is p → q;*0668

*then q → r--this q → r; then p → r is also true.*0674

*So, I can come up with a true conditional statement by using this.*0679

*Then, I can say that my p is here; so if M is the midpoint of segment AB, then the segments are congruent.*0686

*And I can also say "then AM is congruent to MB," because this one uses AB, so I can just say AM is congruent to MB.*0728

*I can write it like this, or I can write it like this.*0744

*Here, I used p; all of this is p; and then, the segments are congruent, so that is r; so this was p to r.*0747

*So, since this is true, and this is true, then this is what I can conclude: p → r is also true, by the Law of Syllogism.*0764

*Let's do the next one: If two angles are vertical, then they do not form a linear pair.*0776

*Here is p; then they do not form a linear pair--this is q; this one is p → q.*0787

*Then, if two angles are vertical--look at this one--this is the same as right here; so this one is p.*0801

*Then, they are congruent; this is r; so this is p → r.*0809

*Well, here I have p → q; and the Law of Syllogism says that p → q and q → r have to be true.*0819

*I can't have p → q and then p → r; I can't come up with a true conclusion, because here it is not q → r; it is p → r.*0828

*With this, I can't form a conclusion; so this one is no conclusion.*0840

*Let's do a few examples: we are going to use the laws of logic, the ones that we just learned, the Law of Detachment and the Law of Syllogism.*0853

*And we are going to determine if statement 3, the third statement, follows logically from true statements 1 and 2.*0865

*Based on the first one and the second one, we are going to see if the third one is going to be a true conclusion.*0874

*Number 1: Right angles are congruent--that is the first statement.*0883

*Now, this is not written as a conditional; so if you want, you can rewrite it as a conditional.*0889

*Or you can just remember that this part right here is going to be the hypothesis, and this part right here is going to be the conclusion.*0894

*I will just write out the conditional of "right angles are congruent": "If the angles are right angles, then they are congruent."*0904

*And that is the congruent sign; if angles are right angles, then they are congruent.*0926

*Now, it is easier to see that this is my hypothesis; that is p; and "they are congruent"--this is q.*0931

*This first one was p → q; now, the second statement is "Angle A and angle B are right angles."*0942

*Here we have right angles; now, do we see that?--that sounds familiar to me.*0954

*It is right here; the angles are right angles; so angles A and B are right angles.*0961

*This is p; or we can write it here--p.*0967

*Then, angle A is congruent to angle B: is that the correct conclusion?*0975

*Well, here, if the angles are right angles, and it says that angle A and angle B are right angles, then what?*0982

*They are congruent; so then, this says that they are congruent; so this is q.*0990

*This is true; this is a valid conclusion, based on the Law of Detachment, because the Law of Detachment says that,*0996

*if p → q is true, and p is true, then q is true.*1009

*So, it is valid; this is the Law of Detachment.*1013

*The next one: Vertical angles are congruent.*1023

*Vertical angles: this one is p; them being congruent: that is q, so this one is p → q.*1031

*Angle 1 is congruent to angle 2--now, is that from p or q?*1043

*That is from q, because it says that angles are congruent here; so this one is q.*1051

*And angle 1 and angle 2 are vertical angles--this is p...this is actually...I wrote p instead of q right here.*1058

*This one is q, and the conclusion was that angle 1 and angle 2 are vertical angles, which is p.*1069

*Now, we don't have a law of logic that says that if p → q is true, and q is true, then p is true.*1078

*That is not any law; it looks like the Law of Detachment, but the Law of Detachment is that if p → q is true, and p is true, then q will be true.*1088

*OK, so in this case, this is an invalid conclusion.*1103

*See, p → q and p--then q will be true; it can't be the other way around.*1114

*This is invalid; this is actually the converse, and that is not true.*1120

*Again, using the Law of Detachment and the Law of Syllogism, determine if statement 3 follows logically from true statements 1 and 2.*1131

*And state which law is used.*1140

*The first one: inline skaters live dangerously: here, "inline skaters" would be p; they "live dangerously"--that is q.*1144

*"If you live dangerously"...that is the same thing as q; so this is q..."then you like to dance"; this is a new statement, so this is r.*1162

*"Inline skaters"--this is p--"like to dance"--this is r.*1178

*So, here "inline skaters live dangerously" is p → q; this is q → r; then the third statement,*1188

*"inline skaters like to dance"--this is p → r; and this is valid by the Law of Syllogism.*1199

*This one says, "Inline skaters live dangerously"; that is p → q.*1216

*If you live dangerously--that is the same statement as this one right here--then you like to dance; that is q → r.*1223

*So, this one right here and this one right here are the same.*1232

*"Inline skaters"--that same statement right there is p--"like to dance"--that is r.*1237

*And the Law of Syllogism, remember, says if p → q is true, and q → r is true, then p → r is true.*1244

*So, it is like the transitive property--the Law of Syllogism.*1251

*The next one: "If you drive safely, the life you save may be your own."*1256

*Here, this is p; "the life you save may be your own"--here is q.*1266

*"Shani drives safely"--that is from p; "the life she saves may be her own"--this one is q.*1276

*This is the same as this one; so the first statement is p → q, and the next one is p; the conclusion,*1291

*"the life she saves may be her own," is q; so based on 1 and 2, based on these two, we are able to get this.*1304

*Yes, this is valid; and this is one is by the Law of Detachment.*1314

*The Law of Detachment says that if p → q is true, and p is true, then q is true.*1325

*We are going to do a few more examples: the first one: Draw a conclusion, if possible; state which law is used.*1335

*If you eat to live, then you live to eat: "If you eat to live"--this one is p--"then you live to eat"--that is q; that is p → q.*1344

*"Christina eats to live": that is from statement p, so draw a conclusion.*1362

*Our conclusion is, then, "Christina lives to eat," because if p → q is true, and p is true, then I can conclude that q is true.*1372

*And this one was by the Law of Detachment.*1393

*The next one: "If a plane exists, then it contains at least three points not on the same line."*1404

*"If a plane exists"--there is p--"then it contains at least three points not on the same line"--there is q.*1412

*And to draw this out: this is just saying that if I have a plane, then contains at least three points in the plane that are not on the same line.*1423

*Plane N (let me draw plane N--here is plane N) contains points A, B, and C, which are not on the same line.*1440

*If a plane exists, then it contains at least three points not on the same line.*1466

*Plane N contains points A, B, and C, which are not on the same line.*1471

*Well, all of this right here is from statement Q; we have...so I have p → q, and then I have a q.*1477

*So, I cannot come to a conclusion; I cannot draw a conclusion, because there is no law that says that if p → q is true, and q is true, then p is true.*1507

*So, I can't say, "Then plane N exists"--that is not a conclusion that I can come to.*1521

*In this case, my answer will be no conclusion--it cannot be done.*1527

*Draw a conclusion, if possible; state which law is used.*1541

*If you spend money on it, then it is a business; if you spend money on it, then it is fun.*1546

*Let's label these: this right here is p; "then it is a business" is q; so this is p → q.*1552

*"If you spend money on it"--well, that is p; "then it is fun"--this is r; it is not the same as q, so it is r; this is p → r.*1563

*Now, can I draw a conclusion based on these statements?--no, because there is no law that says that,*1579

*if p → q is true, and p → r is true, then q → is true.*1591

*It has to be p → q; so the Law of Syllogism says p → q and q → r; then p → is true--this is the Law of Syllogism.*1600

*So, in this case, since it is p → q and p → r, this has no conclusion.*1627

*The next one: if a number is a whole number, then it is an integer.*1640

*Remember that whole numbers are numbers like 0, 1, 2, 3, and so on.*1645

*And integers are whole numbers and their negatives, so it is going to be -2, -1, 0, 1, 2, and so on; those are integers.*1656

*"If a number is a whole number"--there is my p--"then it is an integer"--there is a q.*1672

*"If a number is an integer"--isn't this q?--"then it is a rational number"--this is r.*1684

*And rational numbers are numbers that are integers (it could be -2); I can have fractions; I can have terminating decimals--all of that.*1696

*Now, remember: these given statements are true statements, and you are trying to see if you can use those true statements to draw a conclusion.*1714

*Here is p → q; this one is q → r; remember: if we have p → q and q →, then we can say that p → r is true.*1727

*That is going to be our conclusion: p → r; so I can say, "If a number is a whole number, then"--here is p--*1737

*I am going to draw my conclusion, p → r--"then it is a rational number."*1759

*All of this is p, and all of this is r; and that would be valid because of the Law of Syllogism.*1771

*The next example: Determine if statement 3 follows logically from statements 1 and 2; if it does, state which law is used.*1791

*Based on 1 and 2, we are going to see if number 3 is valid.*1800

*If you plan to attend the university of Notre Dame, then you need to be in the top 10% of your class.*1806

*Here is my p; "you need to be in the top 10% of your class"--there is my q; so this one is p → q.*1823

*Jonathan plans to attend Notre Dame; so this one is p.*1836

*Jonathan needs to be in the top 10% of his class; this is q--yes, that is q--"then he needs to be in the top 10% of the class."*1848

*Based on these two, numbers 1 and 2, these are true statements; statement 1 is true, and statement 2 is true.*1861

*Then, is my conclusion, my statement 3, true?--yes, this statement is valid, because of the Law of Detachment.*1868

*The next example: Determine if statement 3 follows logically from statements 1 and 2.*1889

*If it does, state which law was used.*1896

*We are going to see, again, if the third statement is valid or invalid, based on these two true statements.*1899

*So, if an angle has a measure less than 90, then it is acute.*1908

*"An angle has a measure less than 90"--that is my p; then "it is acute"--this is q; so my conditional is p → q.*1915

*"If an angle is acute"--well, isn't that what this is right here?--so here is q.*1927

*"Then its supplement is obtuse"--the supplement is an angle measure that makes it 180.*1936

*So, if we have two supplementary angles, then it is two angles that add up to 180.*1950

*A supplement of an angle would be the number, the angle measure, that you would have to add so that it would add up to 180.*1955

*Then, it is obtuse; this is r; this is the new statement, so here we have q → r.*1963

*"If an angle has a measure less than 90"--here is my p; all of this is p--"then its supplement is obtuse"--this is r.*1972

*So, this, my third statement, was p → r; well, does that follow any rule, any law?*1996

*p → q is true; q → r is true; then p → r is true; so this is valid, and it is from the Law of Syllogism.*2006

*And the next example: If a figure is a rectangle, then its opposite sides are congruent.*2027

*If I have a rectangle, its opposite sides are congruent; so this is congruent to here, and this is congruent to here.*2035

*AB is congruent to DC; so if I have ABCD, AB is congruent to DC, and AD is congruent to BC.*2051

*The figure is a rectangle--there is my p; then its opposite sides are congruent, so there is q.*2067

*AB is congruent to DC, and AD is congruent to BC.*2075

*Here is q, because it says "if the opposite sides are congruent"; ABCD is a rectangle; this is p.*2092

*I am going to use a different color for that one; this is p.*2111

*So, statements 1 and 2 are p → q, and statement 2 is q, and my third statement is p.*2116

*Can you use these two to make this conclusion, that p is true?--no, so this is invalid.*2129

*This statement right here is invalid; the Law of Detachment says, if p → q is true, and p has to be true, then q is true, not the other way around.*2136

*This one is invalid; make sure that this second statement has to be p, and then your conclusion is going to be q.*2148

*OK, well, that is it for this lesson; thank you for watching Educator.com.*2160

3 answers

Last reply by: Khanh Nguyen

Thu Apr 30, 2015 6:55 PM

Post by Edmund Mercado on February 20, 2012

On Using Laws of Logic, cont.: How does it follow logically that if you live dangerously, then you like to dance?