Geometry 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 20335

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 up0526

(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 assume1169

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 parts1733

(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