Geometry 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 lines0319

(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 p0338

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 lines1542

(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 lines2202

(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