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### Proving Lines Parallel

- Postulates:
- If two lines in a plane are cut by a transversal so that corresponding angles are congruent, then the lines are parallel
- 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
- Parallel Postulate: IF there is a line and a point not on the line, then there exists exactly one line through the point that is parallel to the given line
- Theorems:
- If two lines in a plane are cut by a transversal so that a pair of consecutive interior angles is supplementary, then the lines are parallel
- If two lines in a plane are cut by a transversal so that a pair of alternate interior angles is congruent, then the lines are parallel
- In a plane, if two lines are perpendicular to the same line, then they are parallel

### Proving Lines Parallel

If lines m and n are cut by a transversal line q and the corresponding angles are congruent, then line m is ____ parrallel to line n.

Point A is not on line m, then there are more than one lines that pass through point A and are prallel to line m.

- line r and line p are parallel.

- line n is parallel to line l.

^{o}, find a pair of parallel lines, and state which postulate or theorem is used.

- line p is parallel to line q.

- line p is parallel to line r.

- line p is parallel to line q.

^{o}, find m∠3.

- ∠2 ≅ ∠1
- m∠2 = m∠1 = 120
^{o} - m∠2 + m∠3 = 180
^{o}

^{o}− m∠2 = 180

^{o}− 120

^{o}= 60

^{o}.

- ∠2 ≅ ∠1
- m∠2 = m∠1
- 2x + 4 = x + 20

*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 Lines Parallel

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 0:00
- Postulates 0:06
- Postulate 1: Parallel Lines
- Postulate 2: Parallel Lines
- Parallel Postulate 3:28
- Definition and Example of Parallel Postulate
- Theorems 4:29
- Theorem 1: Parallel Lines
- Theorem 2: Parallel Lines
- Theorems, cont. 6:10
- Theorem 3: Parallel Lines
- Extra Example 1: Determine Parallel Lines 6:56
- Extra Example 2: Find the Value of x 11:42
- Extra Example 3: Opposite Sides are Parallel 14:48
- Extra Example 4: Proving Parallel Lines 20:42

### Geometry Online Course

### Transcription: Proving Lines Parallel

*Welcome back to Educator.com.*0000

*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 show*0065

*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 BIK*0604

*(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 relationship*0754

*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 supplementary*0986

*(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 simplified*1160

*(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

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