### Proving Angle Relationships

- Supplement Theorem: If two angles form a linear pair, then they are supplementary angles
- Congruence of Angles: Congruence of angles is reflexive, symmetric, and transitive
- Angle Theorems:
- Angles supplementary to the same angle or to congruent angles are congruent
- Angles complementary to the same angle or to congruent angles are congruent
- All right angles are congruent
- Vertical angles are congruent
- Perpendicular lines intersect to form four right angles

## Proving Angle Relationships

## If ∠ABC and ∠CBD form a linear pair, and m∠ ABC = m∠CBD

Find m∠ABC.

- ∠ ABC and ∠ CBD form a linear pair (Given)
- m ∠ABC + m ∠CBD = 180
^{o}(Definition of linear pair) - m ∠ABC = m ∠CBD (Given)
- m ∠ABC + m ∠ABC = 180
^{o}(substitution property of equality)

### m ∠ABC = 90^{o} (Division property of equality)

## Complete the statement with always, sometimes, or never.

Two angles that are supplementary to the same angle are ____ congruent.

### Always.

## Complete the statement with always, sometimes, or never.

Two supplementary angles are _____ right angles.

### Sometimes.

## Complete the statement with always, sometimes, or never.

Vertical angles are _____ congruent.

### Always.

## Complete the statement with always, sometimes, or never.

Two lines intersect, they _____ form four right angles.

### Sometimes.

## Complete the statement with always, sometimes, or never.

Congruent angles are ______ right angles.

### Sometimes.

##

Line AB and line CD intersect at point O, m∠AOC = 4x + 3, m∠COB = 5x + 6, find m∠AOC and m∠COB.

- ∠ AOC and ∠ BOC are supplementary angles
- m ∠AOC + m ∠COB = 180
- 4x + 3 + 5x + 6 = 180
- 9x = 171
- x = 19
- m ∠AOC = 4x + 3 = 4*19 + 3 = 79

### m∠BOC = 5x + 6 = 5*19 + 6 = 101.

## ∠1 and ∠2 are both complementary to ∠3. m ∠1 = 2x + 3, m ∠2 = 3x + 4, find x.

- ∠1 is congruent to ∠2.
- m∠1 = m∠2
- 2x + 3 = 3x + 4

### x = − 1.

## ∠1 and ∠2 are vertical angles, ∠2 and ∠3 are completary angles, m ∠1 = 25^{o}, find m ∠3.

- ∠1 and ∠2 congruent
- m ∠2 = m ∠1 = 25
^{o} - ∠2 and ∠3 are complementary angles
- m ∠2 + m ∠3 = 90
^{o}

### m ∠3 = 90^{o} − m ∠2 = 90^{o} − 25^{o} = 65^{o}.

## Justify the statement with a property of congruence of angles.

If 1 ≅ 2, then 2 ≅ 1.

### Symmetric

*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

## Proving Angle Relationships

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
- Supplement Theorem
- Congruence of Angles
- Angle Theorems
- Angle Theorems
- Using Angle Theorems
- Example 1: Always, Sometimes, or Never
- Example 2: Always, Sometimes, or Never
- Example 3: Always, Sometimes, or Never
- Extra Example 1: Always, Sometimes, or Never
- Extra Example 2: Find the Measure of Each Angle
- Extra Example 3: Find the Measure of Each Angle
- Extra Example 4: Two-Column Proof

## Mathematics: Geometry

## Transcription: Proving Angle Relationships

*Welcome back to Educator.com.*0000

*This next lesson is on proving angle relationships.*0002

*Let's go over the supplement theorem; this lesson actually uses a lot of theorems.*0007

*And remember, from the last lesson: a theorem is a statement that has to be proved.*0012

*It is not like a postulate, where we can just assume them to be true; the theorems have to be proved in order for us to use them.*0020

*And usually, your book will prove it for you; and then, after that, you can use it whenever you need to.*0029

*The first one (this is the supplement theorem) is "If two angles form a linear pair, then they are supplementary angles."*0035

*Now, supplementary angles are two angles that add up to 180 degrees.*0045

*A linear pair would be two angles, a pair of angles, that form a line--"linear" means line.*0059

*Two angles, a pair of angles, that form a line are supplementary angles.*0070

*That means that if the two angles form a line, then they will add up to 180.*0078

*If angle 1 and angle 2 form a linear pair (here is angle 1, and here is angle 2)--because you put them together, they form a line--*0085

*and the measure of angle 1 is 85 (this is 85), then find the measure of angle 2.*0095

*Then, this is what we are trying to find.*0102

*Well, we know that, since these two angles form a linear pair, they are supplementary; that means that they add up to 180.*0105

*And then, if the measure of angle 1 is 85, well, this plus this one is 180, so I can say 85 degrees, plus the measure of angle 2, equals 180.*0115

*This is the measure of angle 1 that is given; and then, the measure of angle 2...together, they add up to 180.*0130

*If I subtract 85, what will I get here? The measure of angle 2 is 95 degrees.*0138

*That is how I am going to find the measure of angle 2--using supplementary angles and the supplement theorem.*0150

*OK, congruence of angles is reflexive, symmetric, and transitive.*0159

*Reflexive: remember, reflexive is when a = a; that is the reflexive property.*0167

*The symmetric property is just "if a = b, then b = a."*0173

*And then, the transitive property is, "if a = b, and b = c, then a = c."*0178

*So, if you need to review those properties, then just go back a few lessons before when we talked about each of those in more detail.*0185

*But congruence of angles: when we talk about angle congruence, then these properties also apply.*0197

*This is a proof, because this is a theorem, so we have to prove that.*0208

*And we are just going to prove one of these; so in your book, they will actually show you all of the proofs for each of these,*0216

*but for this lesson, we are just going to just prove one of them.*0224

*Let's see, angle 1 is congruent to angle 2: we know that these two are congruent;*0229

*angle 2 is congruent to angle 3, so prove that angle 1 is congruent to angle 3.*0236

*Which property does that sound like? It sounds like the transitive property.*0243

*We are just saying that we are going to prove that the transitive property can be applied to congruent angles.*0251

*#1: For statements, and then reasons on this side...statement #1: Angle 1 is congruent to angle 2, and angle 2 is congruent to angle 3.*0260

*The reason for that, we know, is "Given"; the first one is always given.*0285

*#2: Since we are trying to prove that the transitive property works for congruent angles, I have to first,*0291

*because I know that the transitive property will work for angles that are equal, change this to measure of angle 1 = measure of angle 2,*0304

*and measure of angle 2 = measure of angle 3; and in that case, all I did was change the congruence to the "equals."*0317

*That is the definition of congruent angles.*0327

*And then, the third one: If the measure of angle 1 equals the measure of angle 2, and the measure of angle 2*0335

*equals the measure of angle 3, then I know that I can apply the transitive property to this one.*0341

*So, this is "the measure of angle 1 equals the measure of angle 3," and this would be the transitive property.*0347

*And I know that that is the transitive property.*0359

*I can use the transitive property, because it is the "equal."*0363

*And then, #4: I am trying to prove that angle 1 is congruent to angle 3; angle 1 is congruent to angle 3, and all I did there was to go from equal to congruent.*0367

*So, it is the definition of congruent angles.*0381

*So then, here, this is the proof to show that if you have this given to you, you can use the transitive property.*0388

*You can prove that the transitive property works for the congruence of angles, and that is just all it is.*0401

*Now, you can use these three properties on the congruence of angles.*0406

*Angle theorems: #1: Angles supplementary to the same angle or to congruent angles are congruent.*0416

*Now, these theorems in your book are probably labeled Theorem 2-Something; make sure you don't use that name.*0426

*Don't call it by whatever your book calls it; the only time that you can use the same name is when there is an actual name for the theorem.*0435

*But don't call it by what the book labels it, if it is 2-Something, because all of the books will be different.*0444

*And if there is no name for it (like number 1--there is no name for it; it is just the theorem itself), then you would have to write out the whole theorem.*0453

*But there is a way for you to abbreviate it: Angles supplementary to the same angle, or to congruent angles, are congruent:*0462

*so then, you can just say, "Angles supplementary to same angle or to congruent angles are congruent."*0470

*You can just write it like that; any time you have "angles," you just write the angle sign; if you have "congruent," then write the congruent sign.*0482

*And then, you can just shorten "supplementary."*0491

*Now, what is this saying? "Angles supplementary to the same angle or to congruent angles are congruent."*0496

*Well, if I have an angle, say angle 1; and let's say this angle is supplementary to angle A (this is angle A);*0504

*angle 1 and angle A are supplementary angles--that means that, if you add them up*0526

*(you add the measure of this angle and add the measure of that angle), you get 180.*0532

*Now, let's say I have another angle, angle 2: now, this angle is also supplementary to angle A.*0537

*Then, they are saying that these two angles have to be congruent.*0549

*If two angles are supplementary to the same angle, then they have to be congruent.*0554

*And if this is 100 degrees (the measure of that is 100), then this has to be 80.*0560

*There is only one angle supplement to this, and there is only one angle supplement to that.*0569

*So, if these two are supplementary, and these two are supplementary, the only angle measure that this can be is 100.*0578

*I can't find another angle measure that will be supplementary to this angle that is different than 100.*0588

*As long as they are both supplementary to the same angle, then they have to be congruent.*0598

*You can't have two angles supplementary with different measures.*0602

*That is what this is saying: if angles are supplementary to the same angle (both of these are supplementary to the same angle, A), then they are congruent.*0611

*And the same thing here: Angles complementary to the same angle or to congruent angles are congruent.*0626

*If I have...here is angle 1, and that is complementary to angle A; and then I have angle 2;*0631

*if this is, let's say, 70, and angle 1 and angle A are supplementary, then this has to be 20 degrees, because "complementary" is 90.*0649

*So, if the measure of angle 1 is 70, and the measure of angle A is 20,*0670

*and let's say that the measure of angle 2 is also complementary to angle A;*0675

*then this has to also be 70, so they are going to be congruent; angle 1 is congruent to angle 2.*0679

*More theorems: All right angles are congruent.*0694

*If I have a right angle, the measure of that angle is 90; if I have a right angle like this, guess what that is--90.*0701

*If I have one like this, that is still 90; so all right angles are congruent, because they all have the same measure.*0718

*Vertical angles are congruent: this one is actually going to be used a lot.*0730

*Remember: vertical angles are angles like this one and this one.*0737

*These are vertical angles, and these are vertical angles; so that means that these two, the angles that are opposite each other, are going to be congruent.*0746

*And then, these two angles that are opposite each other are also congruent.*0756

*Vertical angles are always congruent; they are not congruent to each other--make sure that you don't get it confused with this one and this one.*0762

*It is always going to be the opposite, so this and this.*0772

*Perpendicular lines intersect to form four right angles.*0778

*If I have perpendicular lines, this is 90; this one right here (remember, linear pairs are always supplementary)--*0782

*if this is 90, then this one has to also be 90, because they form a line.*0799

*And then, this angle and this angle form a line; that is a linear pair, so this has to be 90; and the same thing here--this has to be 90.*0805

*They all become right angles; as long as the lines are perpendicular, you have four right angles.*0815

*Complete each statement with "always," "sometimes," or "never."*0827

*Two angles that are complementary to the same angle are ___ congruent.*0831

*Two angles that are complementary to the same angle--I have two angles, and they are complementary to the same angle, A.*0839

*This is complementary to A, and angle 2 is complementary to A; then they have to always be congruent.*0856

*Vertical angles are [always/sometimes/never] complementary.*0869

*Vertical angles, we know, are like this; so, vertical angles are complementary...*0873

*well, we know that vertical angles are congruent; that means that,*0884

*if they are to be complementary, then that means they have to add up to 90.*0887

*But then, they have to be the same measure; so if they are going to add up to be 90, then each of them has to be 45.*0894

*In that case, then the vertical angles would be complementary, because they add up to be 90 degrees.*0904

*But look at these angles right here: these angles are not complementary, so what would this angle be?*0911

*This would be 135; how do I know that?--because this is a linear pair, and they are supplementary.*0921

*This one and this one...135 + 45 has to add up to 180, because it is a line; they form a linear pair.*0930

*This angle and this angle are congruent, because they are vertical; vertical angles are always congruent.*0939

*But they are not complementary; so in this case they can be complementary, but in this case they are not.*0948

*So, here, this would be my counter-example--an example that shows that something is not true,*0955

*an example that shows where this statement is not going to be true.*0963

*Then, I know that it could be complementary, and it might not be complementary.*0969

*So, this will be sometimes.*0976

*Two right angles are ___ supplementary; two right angles are [always/sometimes/never] supplementary.*0982

*If I have a right angle, that is 90; if I have another right angle, that is 90; 90 and 90 always make 180.*0992

*So, this will be always; two right angles are always supplementary.*1004

*OK, let's do a few examples: Two angles that are supplementary [always/sometimes/never] form a linear pair.*1015

*Let me think of a counter-example: try to think of two angles that are supplementary that do not form a linear pair.*1028

*Well, how about if I have an angle like this; let's say this is 100 degrees, and then I have another angle like this that is 80 degrees.*1039

*They are supplementary, right? Yes, they are, because they add up to 180.*1057

*Do they form a linear pair? No, so this is my counter-example.*1061

*Can they form a linear pair, though?--yes, because, if I have a linear pair, here is 100 and here is 80.*1075

*So, my answer will be sometimes.*1088

*Two angles that form a linear pair are [always/sometimes/never] supplementary.*1097

*Two angles that form a linear pair--that means that we have to have a linear pair--are [always/sometimes/never] supplementary.*1102

*It is always, because a linear pair will always be supplementary.*1116

*I can make this 120; then this will be 60; if I make this 100, it is going to be 80.*1123

*No matter what, they have to be supplementary, because they form a linear pair.*1129

*Find the measure of each numbered angle: the measure of angle 1 is 2x - 5; the measure of angle 2 is x - 4.*1137

*What do I know about the measure of angle 1 and the measure of angle 2?*1147

*They form a linear pair, so then, they are supplementary; they add up to 180.*1150

*Make sure you don't make them equal to each other, because they are not.*1159

*This is obviously an obtuse angle, and this is an acute angle.*1164

*And they don't look like they are the same; besides that, you just know that you can't always assume*1169

*that two angles that are adjacent are going to be congruent, or that they are going to be equal or have the same measure.*1175

*Keep in mind that two angles that form a linear pair--always remember that a linear pair's angles are going to add up to 180.*1184

*So, I am going to take the measure of angle 1, 2x - 5; I am going to add it to the measure of angle 2: + x - 4.*1192

*And then, I am going to make them equal to 180.*1201

*See how this was the measure of angle 1 and the measure of angle 2, like that.*1205

*And then, I just solve it out; so then, 2x + x is 3x; -5 - 4 is -9; that equals 180.*1218

*I add the 9; it becomes 189; and then I divide by 3, and that is going to give me 63.*1229

*OK, so then, that is my x; and then, what is it asking for?*1242

*They are asking for the measure of each numbered angle; so then, be careful--when you solve for x, you don't leave it like that.*1250

*You have to plug it back in; if they were asking for x, then yes, that would be the answer.*1259

*But in this problem, they are not; they are asking for the measure of the numbered angle.*1266

*Then, you have to plug it back in, because they want you to find the measure of angle 1 and the measure of angle 2.*1271

*So then, for the measure of angle 1, 2x - 5...I have to do 2(63) - 5; and that is going to be 126 - 5; that is 121, so it is 121.*1277

*The measure of angle 2 is 63 - 4, which is 59.*1302

*And then, make sure; you can double-check your answer by adding them up, because if we add them up, then they should add up to 180, and this does.*1309

*So, here is the measure of angle 1, and here is the measure of angle 2.*1320

*Measure of angle 3 and measure of angle 4--their relationship: they are vertical.*1329

*We know that vertical angles are congruent; so in this case, since they are vertical, I can make them equal to each other.*1336

*This is not supplementary, and don't assume that they are complementary.*1347

*They could be supplementary, and they could be complementary, but we don't know that they are.*1352

*What we do know for sure is that they are congruent; they are the same.*1358

*So, I can just make them equal to each other.*1362

*So, the measure of angle 3 is going to equal the measure of angle 4.*1366

*The measure of angle 3 is 228 - 3x = x; that is the measure of angle 3, and this is the measure of angle 4.*1374

*If I add 3x to that side, I am going to get 228 = 4x.*1386

*And then, I divide the 4, I am going to get 57.*1391

*And then, again, they want you to find the measure of the numbered angle, not x.*1402

*So, we take the x, which is 57, and we are going to plug it back in.*1414

*The measure of angle 3 equals 228 - 3(57); and then, 228 minus...this is going to be...171.*1419

*And then, subtract it, and you are going to get 57; and the measure of angle 4 is going to be 57.*1450

*Is that right? Well, what do we know?*1477

*Now, unlike this problem right here, where we can just add them up and then see if they are supplementary,*1480

*because we know that they are supplementary, we can't do that to this, because they are not supplementary; they are congruent.*1484

*We have to check our answer; we have to just see that they have the same measure.*1492

*And they do; so then, that would be the answer for the measure of angle 3 and the measure of angle 4.*1496

*OK, find the measure of each numbered angle; angle 1 and angle A are complementary; angle 2 and angle A are complementary.*1505

*Since we know that this is complementary to this and this is complementary to this, what do we know?--*1519

*that the measure of angle 1 equals the measure of angle 2.*1525

*If two angles are complementary to the same angle, angle A, then they are congruent.*1530

*Angle 1 is complementary to angle A; angle 2 is complementary to angle A; that means that these have to be congruent.*1538

*I can just make these angles equal to each other, and then I just substitute in 2x + 25.*1546

*And then, this is going to be x = 20.*1559

*And then again, I have to look back and see: OK, they want me to find the measure of each number angle,*1567

*so I have to find the measure of angle 1 and the measure of angle 2.*1572

*I have to plug x back in: so the measure of angle 1 equals 3(20) + 5.*1575

*Let's see: here I am going to have 65, and then, even though I know that they are the same value,*1584

*that the measure of angle 1 is going to equal the measure of angle 2, I still want to plug in x and see if I am going to get the same number.*1600

*2 times 20, plus 25--this is 40, and that is 65.*1610

*So then, I do have the same measure, so that shows me that it is right.*1617

*The next example: here, we have a proof; we are going to write a two-column proof.*1628

*Our given statements are that the measure of angle ABC is equal to the measure of angle DEF, or DFE; this should actually be DEF;*1635

*and the measure of angle 1 equals the measure of angle 4.*1656

*We are going to prove that the measure of angle 2 is equal to the measure of angle 3.*1661

*So, remember: my column here is going to be statements; my column here is going to be reasons.*1665

*Remember: my first step is going to be the given statements: so the measure of angle ABC equals the measure of angle DEF.*1680

*And then, the reason for that is going to be "Given."*1694

*Number 2: Let's look at this really quickly, or think about this.*1700

*I have that this big angle is equal to this big angle; they have the same measures.*1709

*And this part, this angle, is equal to the measure of this angle.*1717

*And then, I want to prove that the other part of it is going to be equal to this part.*1727

*Remember: when we have a bigger angle, and we want to break it down into its parts*1733

*(because that is what we are doing: we are dealing with the big angle's parts, and the same thing on this side),*1738

*then we want to use the Angle Addition Postulate, because that is what breaks it down from its whole to its parts.*1746

*The measure of angle ABC equals the measure of angle 1, plus the measure of angle 2.*1758

*The measure of angle 1 plus the measure of angle 2 equals this big thing.*1770

*And the same thing for the other one: the measure of angle DEF equals the measure of angle 4, plus the measure of angle 3.*1773

*And the reason is the Angle Addition Postulate.*1797

*Now, from here, since I know that the measure of angle 1 is equal to the measure of angle 4,*1810

*and I want to prove that these two are equal to each other, I can just make this whole thing equal to this whole thing.*1815

*So, I am going to use step 1, and I am going to replace all of this with its parts.*1825

*This is the formula; this is equal to this--that is what is going on with this one: the measure of angle ABC is equal to the measure of angle DEF.*1835

*I am going to replace the measure of angle ABC with its parts, which is this,*1846

*and then replace all of that with its parts there.*1854

*And the reason is going to be the substitution property--and I can write "equality."*1862

*And then, again, the measure of angle 1 and the measure of angle 4 are equal to each other.*1876

*So, I can just replace one of them with the other; so I just replace this with the measure of angle 4,*1884

*because it is given; and I should actually write that, too, for here, because this is another given statement:*1898

*the measure of angle 1 equals the measure of angle 4.*1906

*Since they are equal to each other, I can just replace one for the other.*1911

*And whenever I do that, that is also the substitution property (I am running out of room).*1917

*Then, here, since these two are the same, I can just subtract it out.*1925

*If I subtract it out, then, the measure of angle 2 equals the measure of angle 3.*1934

*And right here, this is the subtraction property of equality.*1941

*And is that my "prove" statement? Yes, it is, so I am done.*1956

*So, remember to always keep in mind...because, from here, you can go in a lot of different directions;*1962

*you can make a left; you can make a right; you can take any direction; it depends on your destination--where are you trying to get to?*1970

*You are trying to get to this right here; if you are trying to get to this statement right here, you have to lead it from this step to this step.*1979

*Maybe after you write your given statement, or before you even start, just look at it and think,*1991

*"OK, well, if I want to get from here to here, what steps do I have to take--what do I have to do?"*1997

*And once your last statement is the same as this right here, then you are done.*2005

*That is it for this lesson; thank you for watching Educator.com--we will see you next time.*2013

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