Geometry Indirect Proofs and Inequalities

Welcome back to Educator.com.0000

The next lesson is on indirect proofs, and we are going to go over some theories on inequalities.0002

An indirect proof: now, all of the other proofs that we have done until now, like the two-column proof,0013

the paragraph proof, and the flow proofs--those are all direct proofs; those are all starting from point A, going directly to point B,0019

your given statement down to the "prove" statement--those are all direct.0029

Now, this one is an indirect proof, which means that you are going to still prove your statement;0034

you are going to still prove something, but in an indirect way.0042

To write an indirect proof, there are three steps, and the first step is this one right here.0050

The first step is to assume that the conclusion is false--whatever conclusion you have,0060

whatever statement you have, you are either going to state it as false, or you are going to state the opposite.0066

That is all you are going to do: just state the opposite.0081

By stating the opposite, you are going to try to prove that opposite statement.0084

And as you prove the opposite statement, you are going to come across a contradiction of something that you know to be true.0091

So, it could be some kind of fact; it could be a theorem, a postulate...something that we have learned until now.0100

It is going to contradict; you are going to try to prove the opposite statement,0111

and so you are basically proving that the opposite statement is false, and therefore, that the original statement is true.0116

Instead of, like all of the proofs that we have done so far, proving that the original statement is true,0128

you are proving that the opposite of the original is false, and therefore, that the original statement is true.0135

So, that is what an indirect proof is.0141

Again, step 1: You state the opposite of the statement; that is it.0145

Then, step 2: You are going to actually write a reason, trying to prove that the opposite is true.0153

But you won't be able to; this is more like your reasoning--you are showing your steps and your reasons behind why it is true;0163

what is involved in that opposite statement; and then eventually, you are going to come to a contradiction,0176

because obviously, you are showing that the opposite is false, so it is going to come to a contradiction of some known fact or something.0183

Then, you say that, since that statement leads to a contradiction, the conclusion must be true.0193

For example, if I say, "It is sunny outside," if that is my statement, then your step 1 is going to say, "It is not sunny outside."0212

You are stating the opposite, that it is not sunny outside.0225

And then, you can say, for step 2, "Well, it is bright outside; the sun is out; it is shining; it is hot; therefore, all of these show0229

that it is a sunny day"; therefore, it leads to a contradiction, because you stated that it is not sunny.0247

But then, everything that you are saying proves it being sunny; so then, you say, "Therefore, the conclusion must be true--it must be sunny outside."0258

Now, I know that this is a really tough section; so we will try to go through step-by-step,0273

and we will try to get you familiar, or a little more comfortable, with understanding indirect proofs.0282

For these problems right here, we are actually just going to state the first step; we are going to start on the first step.0293

And then, we will work our way through.0299

State the assumption you would make to start an indirect proof.0302

My step 1 for this one (all we are going to do is just the step 1): remember: step 1 is to assume that the opposite is true.0307

So, assume that...2 + 6 = 8, so assume that 2 + 6 does not equal 8.0322

That is the step 1: assume that it is not equal to 8.0337

Then, for number 2, your step 1 here: Assume the suspect is not guilty.0342

That would be the assumption for that one.0361

And then, step 1 for this: this is actually a little bit different--if you say that the measure of angle A is less than the measure of angle B,0368

you can say...let's say, if the measure of angle A is 60, then that means that the measure of angle B would be anything that is greater,0381

because the measure of angle A is smaller than the measure of angle B.0391

So then, I could say it is 70; I could say that it is 61; I can say 100.0394

If I am stating the opposite, that means that I am saying that the measure of angle A is greater than the measure of angle B.0401

But also, because the measure of angle A can only be smaller, that means the measure of angle A and the measure of angle B cannot be the same.0413

That means that, if the measure of angle A is 60, the measure of angle B has to be greater; that means that the measure of angle B cannot be 60.0427

If it is the opposite, then you have to say that the measure of angle A could be greater than the measure of angle B, and it could be equal.0440

Because this one doesn't say that it can be equal, for this one you can say that it can be equal, because it is the opposite.0448

Then, the measure of angle A is greater than or equal to the measure of angle B.0455

OK, let's go over some inequality stuff; now, this is the definition of inequality: For any real numbers A and B,0468

A is greater than B if and only if there is a positive number C such that A = B + C.0478

OK, just this part right here, "if and only if": remember, don't get confused by that; it just means that this statement and the converse are true.0488

Now, we have two numbers, A and B; A is greater than B, so let's say A is 10 (just as an example), and B is, let's say, 6.0507

Well, 10 is greater than B, so 10 is greater than 6.0524

And there is a positive number C such that A = B + C.0528

So, A (10) = B (6) plus C (which is 4); that means that C is 4.0536

And it is just saying that if there are two numbers, B and C, that add up together to get A, then A must be greater than B.0548

This has to be greater than this, because these two together make up 10.0565

So, 10 by itself is going to be greater than 6, the B; that is what it is saying.0575

You can also say that 10 is going to be greater than C: 10 is going to be greater than B, and 10 is going to be greater than C.0584

A is going to be greater than B and C separately.0592

First, let's go over some properties of inequality.0595

The first one is the comparison property; with comparison, you know that you are comparing two things to each other.0599

It could be two or more things, but you are just comparing things with each other--the comparison property.0606

And then, of course, these are all properties of inequality.0611

So, when you are comparing, let's say, A and B, that means that A is greater than B, or you can say A is less than B;0614

you can say A is equal to B--those are used to compare; you are just stating that one is compared to the other.0625

The transitive property: now, we know the transitive property, but then, this is the transitive property of inequality.0635

That just means, let's say, that if the measure of angle A is less than the measure of angle B,0641

and the measure of angle B is less than the measure of angle C, then...0651

and to see what we are going to write as our conclusion, if the measure of angle A is smaller than the measure of angle B,0660

and the measure of angle B is smaller than the measure of angle C...0670

now, just to make this a little easier to see, I can write these angles (angles A, B, and C)0673

according to size: so let's say, since the measure of angle A is smaller than the measure of angle B, that I am going to write0683

angle A small, and then I am going to write angle B bigger, like that.0689

So, that shows that A is smaller than B; and then, B is less than C, so then C has to be even bigger.0695

So then, what can we conclude there? Then the measure of angle A is less than the measure of angle C.0704

That is the transitive property; it is just using the transitive property, but in the form of inequalities.0715

Addition and subtraction properties: now, for this one right here, if I say that A is greater than B, then A + C is greater than B + C.0722

This is the addition property, because you originally had A is greater than B; then you added C to both sides,0747

and that is going to change the statement; it is still a true statement,0755

but you are adding a number to each side, and that would be the addition property.0759

We are really familiar with all of these properties; we all know the addition property and subtraction property.0766

It is just adding something or subtracting something; and the same thing for multiplication and division.0772

It is just multiplying something or dividing something; but you are using these in the form of inequalities.0777

So then, the multiplication property would be like if you have, let's say, 5x > 20.0788

Then, x is greater than 4; so that is a property of inequality, because 5x doesn't equal 20; 5x is greater than 20.0797

And you are using these multiplication and division properties for inequalities.0812

Again, with the comparison property, we are comparing two things to each other, one compared to the other.0820

With the transitive property, if one is smaller than the other, and that is smaller than something else,0827

then the original would be smaller than the measure of angle C.0833

The addition and subtraction properties are the same thing as the addition and subtraction properties of equality; it is just that you are using inequality.0842

And the same happens for the multiplication and division properties.0849

OK, so then, this exterior angle inequality theorem is from the inequality theorem that we went over before,0853

where we said that, if A is greater than B, and then A equals B + C...how we talked about that:0864

because this and this together add up to get A, that means that A is greater than B itself.0886

So, in the same way, this exterior angle inequality theorem applies the same concept.0893

If I have an exterior angle (and if you remember, the exterior angle theorem was when we had0901

an exterior angle of a triangle--so then, if this is, let's say, angle 1, then the exterior angle equals0915

the sum of its two remote interior angles), remember the two remote interior angles:0928

there are three angles of a triangle; now, it is not this one that is next to it, that forms a linear pair;0934

it is the other two angles that are far away from it, that are not touching it--remote interior angles.0943

So, these two would be considered the remote interior angles.0947

If this is A and B, then the measure of angle 1 equals the measure of angle A, plus the measure of angle B.0952

This is the exterior angle theorem; now, the exterior angle inequality theorem says that,0963

if the measure of angle 1 equals this plus this, then the measure of angle 1 is greater than the measure of angle A,0970

and the measure of angle 1 is also greater than the measure of angle B,0984

because again, these two together add up to be the same measure as angle 1.0990

So therefore, angle 1 is greater than each of these by itself.0997

And that is the exterior angle inequality theorem.1003

This is the exterior angle theorem, and then this is the exterior angle inequality theorem.1007

This has to do with the inequalities; and just to read it to you, "If an angle is an exterior angle of a triangle,1012

then its measure is greater...this is supposed to be "than"...the measures of either of its remote interior angles."1022

That means that it is either of its remote interior angles, this one and this one, just by itself.1032

And then, an example would be...let's say the measure of angle 1 is 100; the measure of angle A is, let's say, 55;1042

the measure of angle B is 45; I know that these two...100 = 55 (because the measure of angle 1 is 100,1060

and the measure of angle A is 55) + 45; so A and B together equal 100.1075

Then, since these together add up to be 100, and 100 is greater than each of these1094

(100 is greater than 55, and then 100 is greater than 45), that is what this inequality theorem is saying.1099

OK, let's go over our examples: Draw a diagram for the statement.1113

Angle 1 is an exterior angle of a triangle greater than each of the remote interior angles, 2 and 3.1118

So, basically, you have to draw the same diagram that we drew in the last slide.1129

You can draw any sort of exterior angle; if I have a triangle like this, I can draw an exterior angle right here;1137

so then, the exterior angle is angle 1; there is angle 1; and then, the remote interior angles,1148

2 and 3, have to be away from this angle, in here and in here.1155

You can draw it however you want--like this--as long as you draw it 1, 2, 3.1164

And it is just so that you have some sort of visual, or you understand what it would look like,1172

and understand that this is the exterior angle inequality theorem, and how this angle would be greater than each of these angles by itself.1182

OK, Example 2: Name the property for each statement: If AB < CD, then AB + CD < CD + EF.1198

So then, all we did here is, from the hypothesis statement to the conclusion statement, added the EF's; so this is the addition property of inequality.1210

The next one: if 6x is greater than 30, then x is greater than 5; so what happened here?1228

I divided this, so this is the division property of inequality.1234

If the measure of angle A is less than the measure of angle B (let me just draw this again),1247

and the measure of angle C is less than the measure of angle A (that means that C is really small, like that),1255

then the measure of angle C is less than the measure of angle B.1262

So, this is, we know, the transitive property of inequality.1267

Let's go back to our indirect proofs part of the section; and we are going to work on a few more step 1's.1283

And then, in the next example, we will actually do an indirect proof.1308

The first one: Sally is sick today.1311

Your first step is to say, "Assume that Sally is not sick today."1315

If you were to continue this for steps 2 and 3 of an indirect proof, you can say something like...1337

since this is what you are trying to prove, and you are going to end up proving this false...1344

if you say, "If Sally is not sick today, then she would have come to school today;1350

she doesn't sound so good; so since she didn't come today, therefore she must be sick today."1365

And you can say something about how it contradicts the statement, so Sally's not being sick today is not true, because she is sick.1374

And for your last statement, you can say, "Therefore, Sally must be sick today," or something like that.1384

The exterior angle of a triangle has a larger angle measure than one of its remote interior angles.1390

Then, for this one, you would say, "Assume that the exterior angle of a triangle does not have a larger" (that is how you would say it)1396

"angle measure than one of its remote interior angles."1438

The difference between what is written here, this statement, and the step 1, is to say that it does not.1458

And the next one: 41 is not divisible by 3; now again, you are stating the opposite.1467

You say, "Assume that 41 is divisible by 3."1475

And for this last one, in your step 2, in your reason, you could say, "Well, if 41 is divisible by 3, then there should not be a remainder."1495

"When I divide it, there is a remainder of 2; therefore, this statement is false," or "it contradicts what we know is true."1507

And then, in step 3, you can say, "41 must not be divisible by 3"; then you are stating that this one is true.1524

Just like a direct proof, in the end, you are going to eventually state that it is true.1531

And just to explain indirect proofs again: all of the other proofs that we have done, the direct proofs,1539

are just going straight from the given to the "prove" statement; you are just directly proving from the first step to the last step.1554

It is just that you are proving it.1567

With an indirect proof, you are going to actually state that the given statement is the opposite.1570

You are actually proving that the opposite of the original statement is false, and therefore proving that it is true.1576

Instead of proving that the original statement is true, you are going to prove that the opposite of the original is false.1586

So, it is an indirect way of writing these proofs.1592

Then here, in this problem, we are actually going to write an indirect proof; so let's take a look at this.1597

The given is that lines p and q are not parallel.1604

And then, we are going to prove that the measure of angle 1 is not equal to the measure of angle 2.1611

Now, you can do a regular proof with this; you can do a two-column proof with this.1617

But this is just another type.1621

To do an indirect proof, remember, we are going to do three steps.1627

Step 1: we are going to say that the given is false, or its opposite.1630

We are going to say, "Lines p and q are parallel."1640

And then, you can also write (and it is probably better), "Assume that lines p and q are parallel."1652

My step 2: I am going to start...remember: I am trying to say that the measure of angle 1 is not congruent to the measure of angle 2.1661

Now, these are alternate interior angles; if alternate interior angles are congruent, then we know that these lines have to be parallel.1670

So, I can say, "Angles 1 and 2 are alternate interior angles; if alternate interior..."1683

Or, let's say, "If two lines are cut by a transversal so that alternate interior angles are congruent, then the two lines are parallel."1711

"However"...now, since we said that angles 1 and 2 are alternate interior angles, and that,1772

if two lines are cut by a transversal so that alternate interior angles are congruent, then the two lines have to be parallel,1780

well, doesn't this contradict the given statement?...so, "However, this contradicts the given statement."1790

So, for my step 3, I am going to say, "Therefore, since the assumption" right here in step 1 "leads to a contradiction,1813

the assumption" again, step 1 "must be false, and therefore, the measure of angle 1 does not equal the measure of angle 2."1845

If you were to write your own indirect proof, yours might be a little bit different than mine.1873

All you have to make sure you include is that, since we are assuming that lines p and q are parallel,1879

angles 1 and 2, since they are remote interior angles, must be congruent in order for the lines to be parallel.1889

If the lines are not parallel, that is fine, but the angles will not be congruent.1897

If the two lines are cut by a transversal so that alternate interior angles are congruent, then the two lines are parallel.1901

They must be parallel; but that contradicts the given statement, which is that they are not parallel.1911

So, since this assumption right here, that they are parallel, is a contradiction--it contradicts what we know to be true,1917

this theorem--then the assumption is false; you have to include that this is false. Therefore, this is true.1930

We are explaining it in a way where it contradicts, and then saying, "Therefore, since it is a contradiction, the assumption is false."1950

The measure of angle 1 is not equal to the measure of angle 2.1967

And that is it for this lesson.1978

Indirect proofs are a little bit challenging, but again, all you have to do is just show that this contradicts the given statement.1982

And once you do that, you can say that, since it does contradict, the assumption is false; therefore, this has to be true.1994

Well, that is it for this lesson; thank you for watching Educator.com.2007