Geometry Proving Lines Parallel

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This next lesson is on proving lines parallel.0002

We are actually going to take the theorems that we learned from the past few lessons, and we are going to use them to prove that two lines are parallel.0007

We learned, from the Corresponding Angles Postulate, that if the lines are parallel, then the corresponding angles are congruent.0022

If the lines are the parallel lines that are cut by a transversal, then the corresponding angles are congruent.0035

Now, this one is saying, "If the two lines in a plane are cut by a transversal, and corresponding angles are congruent, then the lines are parallel."0045

So, it is using the converse of that postulate that we learned a couple of lessons ago.0053

And we are going to take that and use it to prove that lines are parallel.0058

Before, when we used that postulate, it was given that the lines were parallel; and then you would have to show0065

that the conclusion, "then the corresponding angles are congruent," would be true.0071

But for this one, they are giving you that the corresponding angles are congruent.0077

And then, the conclusion, "the lines are parallel," is what you are going to be proving.0083

This first postulate: if you look at angles 1 and 2, those are corresponding angles.0090

So, if I tell you that angle 1 and angle 2 are congruent, and they are corresponding angles, then the lines are parallel.0096

As long as these two angles are congruent, then these lines are parallel; so I can conclude that these lines are parallel.0104

Now, again, for them to be corresponding angles, the lines don't have to be parallel.0115

Even if the lines are not parallel, they are still considered corresponding angles.0124

But now, we know that, as long as the corresponding angles are congruent, then the lines will be parallel.0129

The next postulate: If two lines in a plane are cut by a transversal so that a pair of alternate exterior angles are congruent, then the two lines are parallel.0136

So again, this is the alternate exterior angles theorem's converse.0149

The alternate exterior angles theorem said that, if two lines are parallel, then alternate exterior angles are congruent.0156

This is the converse; so they are giving you that alternate exterior angles are congruent; then, the lines are parallel.0168

Depending on what you are trying to prove, you are going to be using the different postulates--either the original theorem or postulate, or the converse, these.0176

If you are trying to prove that the lines are parallel, then you are going to be using these.0188

If you are trying to prove that the angles are congruent, then you are going to be using the original.0191

So, alternate exterior angles are congruent; therefore, we can conclude that the lines are parallel.0197

The Parallel Postulate: this is called the Parallel Postulate: If there is a line and a point not on the line,0209

then there exists exactly one line through that point that is parallel to the given line.0220

What that is saying is that I could only draw one line, a single line, through this point, so that it is going to be parallel to this line.0227

I cannot draw two different lines and have them both be parallel to this line--only one.0239

So, it would look something like that...well, that is not really through the point, but...0246

this is the only line that I can draw to make it parallel to this line.0256

Only one line exists; and that is the Parallel Postulate.0262

Now, we are going to go over a few more theorems now.0270

Before it was postulates, but now these are some theorems that we can use to prove lines parallel.0273

If two lines in a plane are cut by a transversal (this is my transversal) so that a pair of consecutive interior angles is supplementary, then the lines are parallel.0281

Remember consecutive interior angles? They are not congruent; they are supplementary.0292

If you look at these angles, angle 1 is an obtuse angle; angle 2 is an acute angle.0297

They don't even look congruent; they don't look the same.0303

Make sure that consecutive interior angles are supplementary.0307

Again, the original theorem had said, "Well, if the lines are parallel" (that is given), "then we can conclude that consecutive interior angles are supplementary."0312

This one is the converse, saying that the given is that the consecutive interior angles are supplementary.0323

Then, the conclusion is...we can conclude that these lines are parallel.0330

If two lines in a plane are cut by the transversal, so that a pair of alternate interior angles is congruent, then the lines are parallel.0338

So again, if these alternate interior angles are congruent, then the lines are parallel.0350

And make sure that it is not the transversal; it is the two lines that the transversal is cutting through that are parallel.0363

And the next theorem, the last theorem that we are going to go over today for this lesson:0371

in a plane, if two lines are perpendicular to the same line, then they are parallel.0375

See how this line is perpendicular to this line? Well, this is my transversal.0383

So, if this line is perpendicular to this line, and this line is also perpendicular to that same line, the same transversal, then these lines will be parallel.0387

If both lines are perpendicular to the same line, the transversal, then the two lines will be parallel.0405

OK, let's go over a few examples: Determine which lines are parallel for each.0418

This one right here, the first one, is giving us that angle ABC is congruent to angle (where is D?...) DGF.0426

That means that this angle and this angle are congruent.0440

OK, now again, when we look at angle relationships formed by the transversal, we only need three lines.0445

We have a bunch of lines here; so I want to just try to figure out what three lines I am going to be using for this problem,0456

and ignore the other lines, because they are just there to confuse you.0466

This angle right here is formed from this line and this line, so I know that those two lines, I need.0474

And then, this angle is also formed from this line and this line.0480

So, it will be line CJ, line FN, and line AO; those are my three lines.0484

This line right here--ignore it; this line right here--ignore it.0492

We are only dealing with this line, this line, and this line.0495

And from those three lines, we know that this line, AO, is the transversal, because that is the one that is intersecting the other two lines.0499

So, if this angle and this angle are congruent, what are those angles--what is the angle relationship?0509

They are corresponding; and the postulate that we just went over said that, if corresponding angles are congruent, then the lines are parallel.0517

I can say that line CJ is parallel to line FN.0530

The next one: angle FGO, this angle right here, is congruent to angle NLK.0546

So again, I am using this line, this line, and this line, because it is this angle right here and this angle right here.0557

So, for those two angles, their relationship is alternate exterior angles.0567

If alternate exterior angles are congruent, then the two lines are parallel.0575

And those two lines are going to be, since this FN is a transversal, line AO, parallel to HM.0583

And remember that this is the symbol for "parallel."0598

The measure of angle DBI (where is DBI?), this angle right here, plus the measure of angle BIK0604

(BIK is right here) equals 180, so that these two angles are supplementary.0614

Now, these two angles are consecutive interior angles, or same-side interior angles,0621

which means that if they are supplementary (which they are, because 180 is supplementary), then the two lines are parallel.0629

That is what the theorem says; so I know that line AO, just from this information, is parallel to HM.0637

The theorem says that, if the two angles are supplementary, then the two lines are parallel.0654

OK, the next one: CJ is perpendicular to HK; this is perpendicular, this last one.0661

And then, FN is perpendicular to KN; there is the perpendicular sign.0671

Remember the theorem that said that, if two lines are perpendicular to the same transversal, then the two lines are parallel.0677

So then, from this information, I can say that CJ is parallel to FN.0688

The next example: Find the value of x so that lines l and n will be parallel.0703

I want to make it so that my x-value will make them have some kind of relationship, so that I can use the theorem,0715

saying that I have to make two angles congruent or supplementary--something so that I can conclude that the lines are parallel.0728

Let's see: these two angles right here don't have a relationship; this one is an interior angle, and this one is an exterior angle.0742

But what I can do is use other angle relationships; if I use other angle relationships, then I can find some kind of relationship0754

from the theorems or the postulates that we just went over.0770

What I can do: since this angle and this angle right here are vertical angles, I know that vertical angles are congruent.0774

And since vertical angles are always congruent, and these are vertical angles, since this is 4x + 13,0786

I can say that this is also 4x + 13, because it is vertical, and vertical angles are congruent.0792

Now, this angle and this angle have a special relationship, and that is that they are same-side or consecutive interior angles.0801

I know that, if consecutive interior angles are (not congruent) supplementary, then the lines are parallel.0812

As long as I can prove that these two are supplementary angles, then I can say that the lines are parallel.0823

I am going to make 4x + 13, plus 6x + 7, equal to 180, because again, they are supplementary.0832

Then, I am going to solve for x; this is going to be 4x + 6x is 10x; 13 + 7 is 20; 10x = 160, so x is going to be 16.0851

So, as long as x is 16, then that is going to make these angles supplementary, and then the lines will be parallel.0870

So, x has to be 16 in order for these two lines to be parallel.0880

Find the values of x and y so that opposite sides are parallel.0889

OK, that means that I want AB to be parallel to DC, and I want AD to be parallel to BC.0894

So, the first thing is that...now, this is a little bit hard to see, if you want to think of it as parallel lines and transversals.0902

So, what you can do: if you get a problem like this, you can make these lines a little bit longer.0918

Extend them out, so that they will be easier to see.0924

That means that this one right here and this one right here--if this is the line,0939

and these are the two lines that the transversal is cutting through, then these two angles are going to be consecutive interior angles.0947

This angle and this angle are consecutive interior angles.0956

This angle and this angle are also consecutive interior angles, because it goes line, line, transversal.0959

That means that these are same-side interior angles, or consecutive interior angles.0967

Then, I have options: since I have the option of making this one and this one supplementary, I can also say that this one and this one are supplementary.0973

But this one has the y, and this one has an x; so instead of using x and y to make it supplementary0986

(you are going to have 2 variables), I want to use this one first.0996

I want to solve this way, because it has x, and this has x--the same variable.0999

And you want to stick to the same variable.1005

Here I can say 5x (and I am going to use that, if consecutive interior angles are supplementary, then the lines are parallel) + 9x + 12 = 180.1009

And then, if I do 14x + 12 = 180, 14x =...if I subtract that out, it is going to be 168.1028

Then, x equals, let's see, 12; so if x is 12, then that is my value of x, and then I have to find the value of y.1041

Now, this angle measure, then, if x is 12, will be 60, because 5 times 12 is 60.1063

Then, this right here: 9 times 12, plus 12, is going to be 120.1074

Now, I also know that this and this are supplementary, so the 60 + 120 has to be 180; that is one way to check your answer.1082

Now, remember: earlier, we said that this angle and this angle are also consecutive interior angles, which means that they are supplementary.1092

Well, if this is 60, then this has to be 120, because this angle and this angle are supplementary; they are consecutive interior angles.1099

So, since this is 120, to solve for y, I can just make this whole thing equal to 120.1109

7y + 10 = 120; subtract the 10; I get 7y = 110; and then, y = 110/7.1115

And that is simplified as much as possible, so that would be the answer.1139

Now, when you get a fraction, it is fine; it is OK if you get a fraction.1150

You can leave it as an improper fraction, like that, or you can change it to a mixed number.1153

But this is fine, whichever way you do it, as long as it is simplified1160

(meaning there are no common factors between the top number and the bottom number, the numerator and denominator).1165

Then, you are OK; there is x, and there is y.1173

You solved for the x-value and the y-value, so now that these consecutive interior angles are supplementary, I can say that these sides are parallel.1178

And then, since this one and this one, consecutive interior angles, are supplementary, this side and this side are parallel.1192

Now, notice how, for these, I did it once; and then, for these, I had to do it twice,1202

because any time you have the same number of these little marks, then you are saying that they are congruent;1210

if they are slash marks, then they are congruent; if they are these marks, then they are parallel.1222

If it is one time, then all of the lines with one will be parallel.1226

For these, all of the lines with two will be parallel; if you have another pair of parallel lines, then you can do those three times.1233

OK, the last example: we are going to do a proof.1244

Write a two-column proof: before we begin, we should always look at what is given, what you have to prove, and the diagram.1248

Look at it and see how you are going to get from point A to point B.1260

This is what we are trying to prove: that AB is parallel to EF.1268

Angle 1 and angle 2 are congruent--this is congruent; and then, angle 1 is also congruent to 3.1273

So, one time is congruent...two times are congruent.1299

We know that, since these two angles are congruent, and these two angles are congruent,1310

I know that, since these two angles are going to have some kind of special relationship,1317

my theorem and my postulate say that, if they have a special relationship, then the lines are parallel.1323

So, I am going to just do this step-by-step: here are my statements; my reasons I will put right here.1329

Statement #1: We know that we have to write the given, so it is that (let me write it a little bit higher; I am out of room)...1345

#1 is that angle 1 is congruent to angle 2, and angle 1 is congruent to angle 3.1353

And the reason for this is that it is given.1364

#2: Well, if angle 1 is congruent to angle 2, and angle 1 is congruent to angle 3, then I can say that angle 2 is congruent to angle 3.1370

So, I will read it this way: If angle 2 is congruent to angle 1, and angle 1 is congruent to angle 3, then angle 2 is congruent to angle 3.1387

And this is the transitive property of congruency--not equality, but congruency.1396

#1: If a is equal to b, and b = c, then a = c; that is the transitive property, and we are using congruency, so it is not equality; it is congruency.1412

Then, from there, since I proved that angle 2 is congruent to angle 3, I know that they are alternate interior angles.1427

Alternate interior angles are congruent; that means that I can just say that, since alternate interior angles are congruent, then these lines are parallel.1445

Step 3: You can say that angles 2 and 3 are labeled as alternate interior angles.1456

Or, you can just go ahead and write out what the "prove" statement is--what you are trying to prove,1471

since you already proved that they are congruent, and they are alternate interior angles.1476

Depending on how your teacher wants you to set this up, this will be either step 3 or step 4.1481

My reason is going to be: If alternate interior angles are congruent, then the lines...1491

now, this is not the complete theorem, but you can just shorten it...are parallel.1506

And that would be the proof; and make sure (again, since we haven't done proofs in a while)1517

that the given statement always comes first, and the "prove" statement always comes last.1523

It is like you are trying to get from point A to point B.1526

If you are driving somewhere--you start from your house, and you are driving to school--your house is point A, and your school is point B.1530

There are steps to get there; it is the same thing--proofs are exactly the same way.1541

You need to have your steps to get from point A to point B.1545

That is it for this lesson; we will see you soon; thanks for watching Educator.com.1550