### Parallel Lines and Transversals

- Parallel lines: Two lines in a plane that never meet
- Skew lines: Two lines that are not in the same plane and do not intersect
- Transversal: A line that intersects two or more lines in a plane
- Be familiar with exterior angles, interior angles, consecutive interior angles, alternate exterior angles, alternate interior angles, and corresponding angles.

### Parallel Lines and Transversals

The lines seperating lanes on the road.

The top and the side of a box.

A tree in the backyard and the street in front of the house.

Two pairs of parallel planes.

Two pairs of intersecting planes.

Two pairs of parallel segments

Two pairs of skew lines

1 and 5

8 and 9

- 1 and 5 are corresponding angles, transversal is q.

6 and 12

7 and 9

7 and 9 are alternative interior angles, transversal is q.

*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

### Parallel Lines and Transversals

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
- Lines
- Angles Formed by a Transversal
- Interior Angles
- Exterior Angles
- Consecutive Interior Angles
- Alternate Exterior Angles
- Alternate Interior Angles
- Corresponding Angles
- Angles Formed by a Transversal
- Extra Example 1: Intersecting, Parallel, or Skew
- Extra Example 2: Draw a Diagram
- Extra Example 3: Name the Figures
- Extra Example 4: Angles Formed by a Transversal

- Intro 0:00
- Lines 0:06
- Parallel Lines
- Skew Lines
- Transversal
- Angles Formed by a Transversal 4:28
- Interior Angles
- Exterior Angles
- Consecutive Interior Angles
- Alternate Exterior Angles
- Alternate Interior Angles
- Corresponding Angles
- Angles Formed by a Transversal 15:29
- Relationship Between Angles
- Extra Example 1: Intersecting, Parallel, or Skew 19:26
- Extra Example 2: Draw a Diagram 21:37
- Extra Example 3: Name the Figures 24:12
- Extra Example 4: Angles Formed by a Transversal 28:38

### Geometry Online Course

### Transcription: Parallel Lines and Transversals

*Welcome back to Educator.com.*0000

*This next lesson is on parallel lines and transversals.*0002

*Let's go over some lines: first of all, parallel lines are two lines in a plane that never, ever meet.*0008

*We know that, if we have arrows at the ends of these lines, then that means that it is going on forever, continuously.*0024

*And no matter how long these lines are, no matter how far out they go, they will never intersect; they will never meet.*0030

*They are parallel; and the symbol for parallel would be two lines like this.*0039

*So, if I want to say that line AB is parallel to line CD, then I can say that line AB is parallel to CD; and that is how you can say it.*0046

*So then, this symbol right here is for parallel.*0056

*Also, when you have two lines drawn like that, to show that they are parallel, you can draw little arrows like that on the lines.*0063

*And that shows that these two lines are parallel.*0074

*If I want to draw two other lines that are parallel to each other, but not parallel to these,*0076

*then instead of drawing them one time (because, if I draw it one time, that means that all three of these would be parallel),*0084

*only the ones that I have drawn once are parallel together; and then, this one I have to draw twice.*0091

*And that shows that if I drew two for this one and two for this one, then these two are parallel.*0098

*If these two are not parallel, then you could just either not draw them or, to show that they are not parallel,*0105

*this could be drawn twice, and this you can draw three times.*0112

*And then, you know that those would not be parallel, because it has two, and this one has three.*0116

*The next one: skew lines are two lines not in the same plane that will never intersect.*0123

*So, they are like parallel lines in that they don't intersect; that is the only thing that they have in common.*0131

*But they are not in the same plane.*0136

*If you hold two pencils like this--you have a pencil like this, and you have a pencil like this--look: they are not in the same plane, and they do not intersect.*0142

*So, these are skew lines--two lines that would never intersect, but they are not parallel.*0152

*They are not parallel; they do not intersect, because they are not in the same plane.*0160

*I can also draw this; if I draw a box, then I can say (let me draw this) that maybe this side and this side are skew, because they are not in the same plane.*0166

*There is no way that I can draw a single plane that will contain those two lines.*0193

*So then, those would be skew lines; another pair of skew lines would be maybe this one right here, the back one, and this one.*0204

*These two will never intersect, and they are not in the same plane.*0213

*Skew lines that are not in the same plane do not intersect, and they are not parallel.*0216

*Transversal: a transversal is a line that intersects two or more other lines in a plane.*0224

*It doesn't matter if these two lines are parallel or not.*0236

*Let's say that these two lines are not parallel; they are just two lines.*0241

*And a line that intersects both of them, two or more, is a transversal; so the red line is the transversal.*0245

*And that is...we are going to be using transversals in this lesson.*0261

*Here are two lines, and they could be parallel; they don't have to be parallel, so don't assume that they are parallel.*0272

*Even if they are drawn side-by-side, even if they look parallel, do not assume that they are parallel unless they tell you.*0280

*And then, this line right here would be the transversal, because it is the line that is intersecting two or more lines.*0287

*When a transversal intersects two or more lines, there are these angles that are formed.*0298

*Now, the angles, when we have a pair of them--they form relationships.*0306

*And these relationships have names; special angles have special names.*0313

*Now, within the two lines that the transversal is intersecting, we look at the two lines*0319

*(that is this one and this one), and if I label these lines and say that this is l, p, and let's say q,*0326

*and lines l and p are intersected by the transversal; all of the angles between the two lines l and p*0338

*are interior angles, because the angles are inside the two lines; they are within the two lines, in between.*0346

*The interior angles would be angle 3, angle 4, angle 5, and angle 6; they are all interior angles, because they are inside the two lines.*0354

*Exterior angles are all of the angles that are outside the two lines: angle 1, angle 2, angle 7, and angle 8.*0369

*Exterior, exterior; 3, 4, 5, and 6 are all interior; and then these two are exterior.*0382

*Then these four are the actual special names based on the relationships between the angles.*0390

*We know that numbers 2 and 3 are vertical angles; 2 and 4 are adjacent angles, or they could be a linear pair.*0397

*They are also supplementary angles (just a review of those relationships there).*0409

*But when we compare an angle from this part to this part, that is when we have the special names.*0415

*The first one: consecutive interior angles are going to be..."consecutive" just means that they are right next to each other, so the one right after.*0426

*And they are interior angles: these two words will mean something--we know that interior angles are angles in between the two lines, l and p.*0443

*4 and 6 are consecutive (it just means that they are next to each other--this angle 4 and this angle 6 are next to each other).*0460

*That means they are the angle right after each other; and it is all on the same side of the transversal.*0468

*So actually, another name for this is "same-side interior angles."*0475

*Some textbooks will call it "same-side interior angles"; but the main name that it is mostly called is "consecutive interior angles."*0490

*Angles 4 and 6, and the other angles would be 3 and 5...these have to be paired up.*0502

*It is a relationship between the two angles; so angles 4 and 6 are consecutive interior angles, and then 3 and 5 would be consecutive interior angles.*0522

*They have to be consecutive, one right after the other, angle 3 and angle 5.*0534

*And they are inside the two lines; 3 and 5 is one pair of consecutive interior angles, and angles 4 and 6 are another pair of consecutive interior angles.*0541

*The next one: alternate exterior angles: "alternate" means that they are alternating sides of the transversal.*0554

*Again, remember: these are pairs; it is a pair of angles that is called alternate exterior angles.*0563

*So, if one angle is on the right side of the transversal, then the other angle has to be on the left side of the transversal.*0568

*So, here is a transversal; we know that this is a transversal.*0577

*It doesn't matter which one, if it is left/right or right/left; it doesn't matter.*0589

*But then look: "exterior angles"--that means that it is on the outside of the two lines--not the transversal, but the other two lines.*0594

*So then, always for these special types of angle relationships, there are always at least three lines involved,*0603

*the two lines and the transversal that is cutting through both lines--at least three.*0613

*Alternate exterior angles are like angle 2 and angle 7, because one is on the right side of the transversal,*0621

*and one is on the left side of the transversal, but then they also have to be on the outside of these two lines.*0632

*So then, angle 2 is outside and on the right; angle 7 is outside and on the left;*0640

*so alternate exterior angles would be like angle 2 and (and that is just an and sign) angle 7.*0647

*Also, another pair would be angle 1 and angle 8.*0657

*Angle 2 and angle 7, and angle 1 and angle 8: there are two pairs.*0670

*Now, that is alternate (left/right; right/left) exterior angles.*0676

*The next one is alternate (again, meaning left/right, right/left of the transversal) interior angles.*0683

*Interior means that it has to be within the two lines; so if I have, let's say, angle 4 and angle 5, they would be alternate interior angles,*0693

*because they are alternating; one is on the right, and one is on the left of the transversal;*0706

*and they are both interior; they are both on the inside of the two lines.*0711

*So, angle 4 and angle 5...and then the other pair would be angle 3 and angle 6; angle 4 and angle 5, and angle 3 and angle 6.*0716

*Corresponding angles: now, here we don't have the words "consecutive" or "alternate," and we don't have the words like "interior" or "exterior."*0736

*Corresponding angles are a little bit special; this one is typically the one that students get a little confused with.*0748

*But just think of it as the angles on the same side of the intersection.*0759

*Let's pretend that this right here is an intersection, like a street intersection.*0768

*And let's say angle 2 is on the top right corner of the intersection; then, from this intersection, the same position would be corresponding angles.*0774

*So, the top right is angle 2; the top right is angle 6; so angle 2 and angle 6 are corresponding angles.*0790

*From the two intersections, if this is one intersection and this is another intersection,*0799

*then it is the two angles that are in the same position, the same location, located in the same top right corner, bottom left--whatever it is.*0803

*So, for angle 1, what is corresponding with angle 1?*0816

*Well, angle 1 is in the top left corner of the intersection; angle 5 is in the top left corner of this intersection.*0821

*So, angle 1 and angle 5 are corresponding angles.*0828

*Angle 1 and angle 5, and then angle 2 and angle 6, angle 3 and angle 7, and angle 4 and angle 8--those are corresponding angles.*0833

*Again, we have exterior angles, angles in the outside of the two lines; interior angles, all of the angles inside;*0860

*consecutive, or same-side, interior angles, would be like 4 and 6--they both have to be on the inside and on the same side;*0867

*so if one is on the right, the other one has to be on the right; or if it is on the left, then it has to be on the left.*0881

*4 and 6 are consecutive interior angles; 3 and 5 are consecutive interior angles.*0889

*Alternate exterior angles, like angle 2 and angle 7, have to be alternating and on the outside; angles 1 and 8 are alternate exterior angles.*0895

*3 and 6 are alternate interior angles.*0908

*Corresponding angles are the angles that are located in the same corner of each of the intersections, so angle 2 and angle 6, or angle 3 and angle 7.*0912

*Those are the special angles formed by a transversal.*0924

*OK, so then let's name the relationship between each pair of angles.*0931

*Angle 2 and angle 7 (let's write these out); angle 1 and angle 6 (I am just going to write the angle signs on that)...*0935

*OK, angles 2 and 7: well, look at these lines really quickly.*0950

*We have this line; let's label this line p and label this line q, and then we can label this line, say, s.*0961

*We know that the only line that intersects two or more lines is line s, this line right here.*0978

*So, that has to be the transversal; these are the two lines that are intersected by the transversal.*0984

*So, when we look at this, even though it looks a little bit different than the diagram from the previous slide,*0991

*we know that these four angles are the interior angles, because the two lines are p and q.*0998

*It is not the transversal, because there is no line next to the transversal that can block in angles and have angles on the outside of them.*1006

*2, 4, 5, and 7 are all interior angles; 1, 3, 6, and 8 are all the exterior angles.*1018

*So then, angles 2 and 7--well, we know that they are inside, and they are alternating sides of the transversal.*1025

*So, this is alternate, and then interior angles.*1036

*Angle 1 and angle 6: now, they are on the same side; they are both above line s, so they are on the same side.*1050

*But they are on the outside, so they are on the exterior.*1068

*Now, there is no relationship that is titled "same-side" or "consecutive exterior angles."*1071

*There is no relationship like that, so in this case, since we don't have a special name for that, we would just say that they are exterior angles.*1081

*They are just exterior angles; they are both on the outside.*1089

*Angle 4 and angle 8: if you look at this, if this is an intersection and this is an intersection, they are both on the same corner.*1096

*4 is on the bottom right; 8 is on the bottom right; so these are corresponding angles.*1108

*Angles 1 and 8 are alternating, so alternate exterior angles; they are alternating, and they are on the outside.*1124

*Angles 2 and 5: they are on the same side, and they are in the interior, so these are consecutive interior angles.*1142

*Or you can call them same-side interior angles, depending on what your textbook calls it.*1158

*So, let's go over a few more examples: Describe each as intersecting, parallel, and skew.*1167

*Intersecting, we know, is when two lines meet; parallel is two lines in a plane that do not meet;*1175

*skew lines are lines that are not in the same plane, and they do not intersect, and they are not parallel.*1182

*The first one, railroad tracks: Railroad tracks, we know, go like this.*1191

*Or you can look at the lines that go this way; either way, these two lines are parallel lines.*1201

*Floor and wall of a room: If we have a floor, and we have the wall, they are intersecting; they are like intersecting planes.*1215

*Number 3: The front and side edges of a desk--if I have a desk, the front and the side edges are intersecting--they intersect right here.*1241

*The flagpole in a park and the street you live on are never going to intersect;*1263

*the flagpole is inside a park that is going this way, and your street that you live on is going this way, so they are skew.*1273

*The next example: Draw a diagram for each: Two parallel planes.*1298

*Now, we talked about parallel lines; planes can also be parallel, as long as they don't ever meet--they don't intersect.*1304

*I can draw a plane like this, and then maybe a plane like this; they won't ever meet.*1315

*Two parallel lines with the plane intersecting the lines: we have a pair of parallel lines, and then,*1328

*if I have a plane (now, I am a horrible draw-er, but let's just say that the plane goes like this,*1342

*where this meets right here and this meets right here), you can say that the plane is intersecting the lines.*1357

*And these are parallel, so I am going to show that they are parallel, like that.*1371

*Two skew lines: now, if I just draw two lines that look skew, since I am drawing on a flat surface,*1375

*it looks like these are going to intersect, because they look like they are on the same plane.*1388

*So, to draw skew lines, the best way to draw skew lines would probably be to draw a box, like a cube.*1395

*And then, I can say that maybe this side right here, and then this side right here, are skew.*1414

*Or I can say (probably a better picture would be) this side right here and this front side right here, the bottom part.*1426

*So then, this line and this line are skew lines, because they are not in the same plane, and then they will never intersect.*1443

*The next example: Name each of the following from the figure.*1454

*Number 1: Two pairs of intersecting planes: Let's say plane CABD and plane...if this right here is on a plane, I can say plane AEB.*1458

*Those two are intersecting, so I can say that those two planes are intersecting.*1480

*Even though planes are usually shown by a four-sided figure, and this is only three, we can just say that this will be on that plane.*1486

*So, I can name a plane by three of the points that are non-collinear: E, A, and B are three non-collinear points,*1497

*so I can just label a plane by AEB; so then, this bottom plane right here is ACBD; that and plane AEB can be two planes that are intersecting.*1505

*Another pair--it could be any of the other ones; they are all intersecting.*1518

*You can say any of the two sides, unless it is the front and the back.*1523

*The front and the back are not...actually, they are intersecting right here; so any of the two planes on here works.*1531

*All pairs of parallel segments: OK, well, we know that parallel segments, or parallel lines*1542

*(segments are just like lines, but they have two endpoints; they don't go on and on forever and ever;*1550

*they have an endpoint and stop, so these would be considered segments instead of lines)...*1556

*now, we know that these segments right here are not parallel, because they intersect; segment AE intersects with segment BE;*1567

*so, those are not parallel segments; all of these that are going up all intersect together, so they are not parallel.*1577

*The only pairs of parallel segments would be CD; I can say that CD is parallel to AB; I can also say that AC is parallel to BD.*1586

*Those are the pairs of parallel lines: these two are never going to intersect, and these two are never going to intersect.*1606

*This one right here can just be plane ABD, or ACBD, and plane AEB.*1616

*Number 3: Two pairs of skew lines--so then, skew lines, I know, are two lines that are not on the same plane and do not intersect.*1636

*I can say that AC and (and don't write this symbol right here, because that means "parallel") BE, right here, are never going to intersect.*1649

*Or I can say that CD and AE are skew lines.*1668

*And all intersecting planes are all of the sides--the planes that contain the sides of these.*1682

*So, I can say plane AEB, plane BED, plane DEC, and plane CEA; those are the four planes.*1689

*All right, the next one: Identify the special name for each angle formed, and state the transversal that formed the angles.*1718

*OK, let me just label these lines and say that this is l; this will be k; this can be n, and this will be p.*1726

*Those are the four lines; that way, when you name the transversal, then you can just name it by the line.*1745

*So, 6 and 09 here is 6, and here is 15.*1756

*Well, now, we have four lines here; so first of all, remember: to figure out special relationships between these two angles, we only need three lines.*1767

*We have four here; so for each of these, you have to be able to identify which lines you are using and which lines you are not.*1782

*And whatever lines you are not using, ignore it or cover it up so that it doesn't confuse you.*1793

*If we are looking at angle 6 and angle 15, then I am looking at this line right here,*1801

*because angle 6 is formed from this line, line n, and line k.*1808

*Those are the two lines involved; and then, remember, there are 3 lines, angle 15 has line p involved.*1814

*For the first ones, 6 and 15, we know that n, p, and k are involved.*1823

*Now, line l is not involved for this pair of angles; so I can cover the whole side up, or just ignore it--*1829

*pretend that this line, line l, doesn't exist--that it is not there, because that is going to confuse you.*1842

*We are just going to look at this part right here; and from this part, angle 6 and angle 15, the line that they have in common is k.*1848

*So, k is the transversal, because it is the line that is intersecting two or more lines, lines n and p.*1863

*Line k is intersecting lines n and p.*1872

*OK, line n would be a transversal, but only when dealing with an angle that involves one of these and one of these.*1875

*Then, line n would be considered the transversal, because that would be the common line between one of those sets of angles.*1884

*But for this problem, we are looking at angles 6 and 15; so then, line k would be the transversal.*1894

*For 6 and 15, they are alternating sides; one is on the right, and one is on the left, of k.*1903

*And they are on the exterior, because 7, 8, 13, and 14 are on the inside; they are all interior.*1909

*6 and 15 are the exterior angles; so this would be alternate exterior angles.*1917

*And then, the transversal is going to be k.*1930

*9 and 13: look at 9, and look at 13; angles 9 and 13 involve line l, line p, and line k, but no line n.*1941

*Line n is not involved in this one; so we ignore line n.*1960

*Cover it up if it is confusing; make sure that you don't look at it--nothing for these two angles involves line n.*1965

*So then, if we are looking at 9 and 13, the transversal is going to be line p.*1975

*9 and 13 would be corresponding angles, because, remember: they are on the same corner of these intersections.*1984

*9 is the top left; 13 is the top left; so these are corresponding (I will just write out the whole thing) angles.*1994

*And then, the transversal would be line p.*2005

*4 and 5: again, these two angles are not involving line p, because angle 4 is formed from lines l and n;*2014

*5 is formed from k and n; so p is ignored.*2028

*So then, the transversal of 4 and 5 would be n, because n is the line that is intersecting both of these lines, the other two lines involved.*2034

*So, from the other two lines that are not the transversal, the inside would be these four right here: 2, 4, 5, and 7;*2045

*4 and 5 are alternating, because one is on the bottom side, and 5 is on the top side.*2054

*And they are on the inside; so it is alternate interior angles.*2061

*And then, the transversal is going to be n.*2072

*3 and 9 are right here; again, this line, this line, and this line are involved, but no k.*2080

*And these are consecutive interior angles, because they are consecutive (or same-side), and they are the inside angles.*2093

*And the transversal between 3 and 9 is going to be line l, because that is the line that is intersecting both of these lines.*2112

*1 and 6 involve this line, this line, and line n, with line n being the transversal, because that is the one that intersects both of these lines.*2121

*And again, these are the same side, but we don't have a name, a special relationship, for same-side or consecutive exterior angles.*2134

*We don't have that one, so these are just going to be exterior angles--they are just both on the outside.*2143

*That is all that it is saying: they are exterior angles, and the transversal is n.*2150

*11 and 14 involve this line, this line, and this line; ignore line n.*2159

*They are alternate, because one is on the bottom, and one is one the top, switching sides of the transversal, p; and they are on the outside.*2169

*So, they are alternate exterior angles, and the transversal is p.*2181

*Again, make sure, when you are looking at special angles that are formed from a transversal,*2196

*that you are only going to be looking at three lines; if you have more than three lines*2202

*(sometimes you might have more than 4; here you only have 4, so there is only one line to ignore;*2207

*other times you might have more)--just make sure that you look at the two angles; look at what lines are involved.*2213

*What is your transversal? Ignore all of the other lines that are there, because it is just going to confuse you.*2222

*You can cover it or just really pretend that it is not there; and do that for each of the problems.*2228

*And that is going to make it a lot easier for you.*2236

*That is it for this lesson; we are going to go over more theorems and postulates in the next lesson on transversals and these special angles.*2240

*Thank you for watching Educator.com; I'll see you soon.*2253

0 answers

Post by Kenneth Montfort on March 5, 2013

can a transversal intersect skew lines?

0 answers

Post by Nagayasu Toshitatsu on December 26, 2012

For extra example III, number 4, does plane ABDC count?