Geometry Deductive Reasoning

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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