### Indirect Proofs and Inequalities

- The three steps to writing an indirect proof:
- Assume that the conclusion is false
- Show that the assumption leads to a contradiction of the hypothesis or something you know is true, like some fact or theorem
- Point out that the assumption must be false, and, therefore, the conclusion must be true
- Inequality: For any real numbers a and b, a > b if and only if there is a positive number c such that a = b + c
- Properties of Inequality:
- Comparison Property: a > b, a < b, a = b
- Transitive Property: If a < b and b < c, then a < c
- Addition and Subtraction Properties: If a > b, then a + c > b + c, and a – c > b – c
- Multiplication and Division Properties: If a > b, then ac > bc, and a/c > b/c
- Exterior Angle Inequality Theorem: If an angle is an exterior angle of a triangle, then its measure is greater than the measures of either of its remote interior angles

### Indirect Proofs and Inequalities

x + 2 = y

The weather is hot and humid.

18 ≥ x + 5

If x + 5 < y + 4, and y + 4 < z − 7, then x + 5 < z − 7.

ACD is the exterior angle of VABC, then which of the following statement is true.

1. m∠ACD > m∠ABC

2. m∠ACD > m∠ACB

3. m∠ACD > m∠A

If m∠1 > m∠2, then 3m∠1 > 3m∠2.

If ―MN ≥ ―PQ , then ―MN −―CD < ―PQ −―CD .

If ―AD < ―EF , then [(―AD )/4] < [(―EF )/4].

The measurement of legs of a right triangle are always smaller than that of the hypotenuse of the right triangle.

Given: ―AD ⊥―BC , ―BD ≠ ―CD

Prove: ―AB ≠ ―AC

- Assume:
- 1. ―AB ≅ ―AC
- 2. m∠ADB = m∠ADC = 90, and ―AB ≅ ―AC , ―AD ≅ ―AD , therefore, ∆ ADB ≅ ∆ ADC (HL),
- So ―BD ≅ ―CD , which conflicts with the given statement
- 3. Since the assumption leads to a contradiction, the assumption must be false, therefore,―AB ≠ ―AC .

2. m∠ADB = m∠ADC = 90, and ―AB ≅ ―AC , ―AD ≅ ―AD , therefore, ∆ ADB ≅ ∆ ADC (HL),

So ―BD ≅ ―CD , which conflicts with the given statement

3. Since the assumption leads to a contradiction, the assumption must be false, therefore,―AB ≠ ―AC .

*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

### Indirect Proofs and Inequalities

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
- Writing an Indirect Proof
- Indirect Proof
- Definition of Inequality
- Properties of Inequality
- Comparison Property
- Transitive Property
- Addition and Subtraction Properties
- Multiplication and Division Properties
- Exterior Angle Inequality Theorem
- Extra Example 1: Draw a Diagram for the Statement
- Extra Example 2: Name the Property for Each Statement
- Extra Example 3: State the Assumption
- Extra Example 4: Write an Indirect Proof

- Intro 0:00
- Writing an Indirect Proof 0:09
- Step 1
- Step 2
- Step 3
- Indirect Proof 4:30
- Example: 2 + 6 = 8
- Example: The Suspect is Guilty
- Example: Measure of Angle A < Measure of Angle B
- Definition of Inequality 7:47
- Definition of Inequality & Example
- Properties of Inequality 9:55
- Comparison Property
- Transitive Property
- Addition and Subtraction Properties
- Multiplication and Division Properties
- Exterior Angle Inequality Theorem 14:12
- Example: Exterior Angle Inequality Theorem
- Extra Example 1: Draw a Diagram for the Statement 18:32
- Extra Example 2: Name the Property for Each Statement 19:56
- Extra Example 3: State the Assumption 21:22
- Extra Example 4: Write an Indirect Proof 25:39

### Geometry Online Course

### Transcription: 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 show*0229

*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 write*0683

*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 had*0901

*an exterior angle of a triangle--so then, if this is, let's say, angle 1, then the exterior angle equals*0915

*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 these*1094

*(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

0 answers

Post by jianhua Ling ling on June 24, 2014

So you change the Given for step 1 not the prove right?