Geometry Angles and Parallel Lines

Welcome back to Educator.com.0000

The next lesson is on angles and parallel lines.0002

OK, last lesson, we learned about the different special angle relationships, when we have a transversal.0007

The transversal with the other lines forms angles, and those pairs of angles have special relationships.0017

And one of them was the corresponding angles.0030

Now, the two lines that the transversal cuts through--remember: I said that the lines can be parallel, but they don't have to be.0040

So, even if the lines are not parallel, you are still going to have corresponding angles.0050

But then, now the postulate is saying that, if the lines are parallel, then the corresponding angles are congruent.0056

If these lines are parallel (let's say that they are parallel lines), then each pair of corresponding angles is congruent--only if the lines are parallel.0068

If we don't have parallel lines--if I have lines like this and like this--they are not parallel;0083

they don't look parallel, but I have a transversal--let's say 1 and 2: these angles are corresponding angles, but they are not congruent.0093

They are not congruent, but they are still corresponding; angles 1 and 2 are corresponding angles, but they are not congruent.0105

They are just called corresponding angles; so be very careful--only if the lines are parallel, then you can see that corresponding angles are congruent.0112

They are the same; they have the same measure; they are congruent.0123

Since these lines are parallel, I can say that angles 1 and 2 are congruent.0127

So, angle 1 is congruent to angle 2; and it goes with all of the pairs of corresponding angles,0137

like this one and this one--they are congruent...this one and this one, and this one and this one.0145

Each of the pairs of corresponding angles is congruent only if the lines are parallel--that is very, very important.0152

And that is a postulate; a postulate, remember, is any statement (such as this) that we can assume to be true.0159

It doesn't have to be proved; if it is a theorem (the next few are actually going to be theorems),0170

then they have to be proved in order for you to be able to use them, because it is not true until it is proven.0176

The next one: here is a theorem; now, we are not going to prove these theorems now, but they are shown in your textbooks.0186

The alternate interior angles theorem--just so you know, some kind of proof has to be shown for the theorems0198

in order for them to be counted as true and correct, and then, that is when we can use them.0208

But for now, since they are proven in your book, we are just going to go ahead and use them.0214

The alternate interior angles theorem says that, if two parallel lines are cut by a transversal0219

(meaning, if the two lines that are cut by a transversal are parallel), then each pair of alternate interior angles is congruent.0225

Again, from the last lesson: if I have two lines...now, I know I am repeating myself a lot,0238

but that is so that you will understand this, because I have seen a lot of students make careless mistakes with these,0245

always thinking that these are congruent; in this case, if I tell you that these lines are not parallel,0258

or if I don't even say anything about them being parallel, then you don't assume that they are parallel.0265

We just have to assume that they are not parallel; then we can't say that angle 1 and angle 2 are congruent.0272

We can say that they are alternate interior angles; that is the relationship; but they are not congruent in this case.0277

So, for lines being parallel (now I am telling you that the lines are parallel), then alternate interior angles0286

(let's say that this is angle 1 and angle 2)...angle 1 is congruent to angle 2.0296

If the lines that are cut by a transversal are parallel, then alternate interior angles are congruent; and that is the theorem, the other one.0306

The next one is the consecutive interior angles theorem: If two lines that are cut by a transversal are parallel,0317

then (this is the tricky part--not tricky, but this is the part that students really make mistakes on) the consecutive interior angles0331

are not congruent; they are supplementary--this is very important.0344

Consecutive interior angles, we know, are angles that are on the same side, like these two angles right here.0351

And they are both on the inside, the interior.0357

So, angles 1 and 2 are consecutive interior angles; but then, they are not congruent--they are supplementary.0362

Only if the lines are parallel, then consecutive interior angles are supplementary.0369

See how the other ones that we just went over are congruent: these are not congruent--they are supplementary.0375

You have to say that the measure of angle 1, plus the measure of angle 2, equals (supplementary means) 180.0383

That means that this angle measure, plus this angle measure, equals 180--very important.0391

And the next one: Alternate exterior angles, if the lines are parallel, are congruent.0404

So, here is a pair of alternate exterior angles; angle 1 is congruent to angle 2.0415

And that also works for this pair of alternate exterior angles, like 3 and 4; those will be congruent, also.0425

Here we have parallel lines that are cut by a transversal.0442

If AB (let's say that this is A, and here is point B--and these are the points, not the angles;0445

here is point C and point D...then AB is a line, so it is line AB) is parallel to line CD,0460

and line CA is parallel to line DB (and then I am going to add these parallel markers;0471

that means that these two are parallel lines, and then for these--this is another pair of parallel lines,0484

so that means that I have to draw two of them for these, because it is another pair), find the values of x and y.0488

So then, here we have 80; and then I need to take a look at x.0500

If I look for a relationship between this one and another one, even though these two have a relationship,0508

this has a variable x, and this has a variable z.0517

I would rather use this relationship, 4x and 80, because, if I am going to compare them, at least this one doesn't have another variable.0521

So, it is easier to solve; so then, if I look at these two, I am only dealing with this line, line AB, line CD, and line BD.0530

That means that line AC, I am going to ignore, because it is not involved in this pair of relationships.0543

Remember from the last lesson: you look at the pair, and when you have the special pair, it only has three lines involved.0549

It only has line AB, line CD, and line BD involved; the other lines that are there--cover them up.0560

Those lines are there for another pair of relationships, so just cover it up.0567

You don't need this line for this pair, so just ignore it.0573

And then, to solve it, the theorem (and the relationship between these two: they are alternate interior angles,0583

because BD would be the transversal between these two lines) says that if the two lines0596

cut by a transversal are parallel (which they are--we know that because it gives us that in here),0604

if the lines are parallel, then alternate interior angles are congruent.0611

Since the lines are parallel, I can say that these two angles are congruent.0618

Then, they are congruent, so 4x = 80; and I divide by 4: x = 20.0621

There is my x-value; and then, for my y, let's look at this one.0638

Well, with this one, I know that, since I have an 80 here, 80 is also congruent to this angle right here,0651

because they are corresponding, and I know that these two lines are parallel.0667

If these two lines are parallel, here is my transversal; that means that this angle right here and this angle right here are corresponding.0670

And as long as the two lines that are cut by the transversal are parallel, then corresponding angles are congruent.0680

So then, I can just write an 80 in here; and then, between this and this, they are vertical.0685

Now, I could have just done this angle right here to this right here; so there are many ways to look at it.0694

You can look at corresponding angles; if you didn't really see the alternate exterior angles--0700

if that is kind of hard for you to see--then you can just say that, OK, they are corresponding, and then these two are vertical.0707

And vertical angles, remember, are always congruent.0711

So, you can say that these two are the same, because they are vertical.0717

Or you can say that this and this are the same, because they are alternate exterior angles.0722

And those are the same, as long as the two lines are parallel.0727

So, either way: 4y + 10 = 80; then 4y = 70; so y = 35/2.0730

And that is just 70/4, and then you just simplify it to 35/2.0755

Now, it doesn't ask for the value of z, but let's just go ahead and solve it.0765

We know that 6z and 80 have a relationship.0774

Now, I know that this is 80, because we found x; x is 20; and 4 times x is 80;0783

and also because they are alternate interior angles, so whatever this is, this has to have the same measure.0790

So, either way we look at it: we can look at it as 6z with this one right here,0798

or we can look at this one with this one right here--same relationship, same value,0801

which also means that this one is also the same as 4x; this angle and this angle have the same measure.0806

Either way, the 6z with this angle right here are consecutive interior angles, or same-side interior angles.0814

Now, if the lines are parallel (which they are), then consecutive interior angles are supplementary--not congruent, but supplementary,0826

which means that I can't make them equal to each other.0839

Consecutive interior angles are the only ones that are not congruent from the special pairs of angles.0844

Supplementary--that means that I have to make 6z + 80 equal to 180.0851

6z = 100; z = 100/6, and then I can just simplify this to 50/3, and that is it; that is z.0859

OK, the last theorem from this section in this lesson is the perpendicular transversal theorem.0897

Perpendicular, we know, are two lines that intersect to form a right angle.0904

So, if I have a line like this and a line like this, and they form a right angle, then they are perpendicular.0911

But then, here we have a transversal involved; so in a plane, if a line is perpendicular to one of the two parallel lines, then it is perpendicular to the other.0918

Here are my parallel lines; I am going to show it by doing that.0932

If my transversal, which is this line right here, is perpendicular to just one of the lines0936

(it doesn't matter which one), as long as these lines are parallel (they have to be parallel),0943

if it is perpendicular to one of the lines, then it has to be perpendicular to the other line.0951

If this is perpendicular to this line, then it is going to be perpendicular to this line, as well.0960

And that is the perpendicular transversal theorem.0967

Now, if the two lines are not parallel (let's say like this), and then I tell you that this line is perpendicular to this line,0969

it is not going to be perpendicular to this line, because these lines are not parallel.0983

In this case, don't assume that it is perpendicular to both--that is only if the lines are parallel.0990

Let's do a few examples: State the postulate or theorem that allows you conclude that angle 1 is congruent to angle 2.0999

Now, remember: the only postulate was the corresponding angles one: that is the one where you have the angles in the same corner,1007

in the same position, in the same corner of the intersection--that is the corresponding angles postulate.1015

Everything else--the consecutive interior angles theorem, the alternate interior angles theorem,1026

the alternate exterior angles theorem--those are all theorems; so the only one is the corresponding angles postulate.1034

Here, what postulate or theorem allows you to conclude that angle 1 is congruent to angle 2?1043

We know that this is our transversal line, because it is the one that cuts through two or more lines.1051

Then, angle 1 and angle 2 are alternate exterior angles.1056

Now, if these two lines are parallel, then we can conclude that angle 1 is congruent to angle 2; let me show that these two lines are parallel, too.1063

Then, this would be the alternate exterior angles theorem.1073

And this one right here--we know that these are corresponding angles.1097

And the only way that the postulate will make them congruent (the only way we can apply the postulate) is if these two lines are parallel, which they are.1106

So, I can say that, by the corresponding angles postulate, angle 1 is congruent to angle 2.1114

All right, the next one: In the figure, line e is parallel to line f.1134

So, let me show this; it doesn't matter which way--I can just do like this, or I can just do like that.1142

AB is parallel to CD, so this one is parallel to this; and the measure of angle 1 is 73.1149

I am going to write that in blue; so this is 73, right here.1158

Find the measure of the numbered angles.1165

All of the numbered angles is what it is asking for.1169

Let's look at this: to look for the measure of angle 2, I know that angle 1 and angle 2 are supplementary, because they are a linear pair.1175

They form a line, and a line is 180 degrees.1190

So, all linear pairs are supplementary; so since linear pairs are supplementary, and these are a linear pair,1196

I can say that 73 plus the measure of angle 2 equals 180.1208

And then, to find the measure of angle 2, I have to subtract the 73; so the measure of angle 2 equals 107.1218

And then, the next one: the measure of angle 3--well, if you look at this, we know that these two lines are parallel.1235

This line intersects both of the parallel lines; so this is a transversal--this line segment AB is a transversal,1246

which means that angle 2 and angle 3 are alternate interior angles.1256

And by the alternate interior angles theorem, since the lines are parallel, we know that these angles are congruent.1263

Since the measure of angle 2 is 107, I can say that the measure of angle 3 is 107.1272

And then, the measure of angle 4: it is also alternate interior angles with angle 1, so by that theorem, again,1284

since the two lines are parallel, those two will be the same; so it is 73.1303

Then, the measure of angle 5 is corresponding with angle 5; angle 5 and angle 1 are corresponding,1314

because it is as if I extend this line segment, just to help me out here: these two lines are parallel;1324

here is my transversal; can you see that?--this is a line, and this is a line; here is that transversal;1332

angle 1 and angle 5 are corresponding, so if this is 73, then the measure of angle 5 has to be 73.1344

And then, the measure of angle 6--you can say that angle 6 is also corresponding with angle 3.1354

So, if you extend this out again, there is my intersection, angle 3, and then my intersection, angle 6.1368

The measure of angle 3 is 107, so the measure of angle 6 is also 107.1378

Angle 7 is alternate interior angles with angle 6, so that has to be the same, since the lines are parallel.1386

And the two lines involved would be this line and this line--can you see that?--this line and this line, and here is my transversal.1399

These two lines are parallel, so angle 6 and angle 7 are congruent by the alternate interior angles theorem.1408

And then, the last one, the measure of angle 8: it is supplementary with angle 6, because it is a linear pair.1421

Or it is alternate interior angles with angle 5, or it is corresponding with angle 4; there are a lot of different relationships going on here.1431

If you want to use the alternate interior angles theorem with angle 5 and angle 8, then it is going to be 73.1447

If you want to look at the corresponding angles postulate with angle 4, then it is also 73.1457

If you want to say that it is supplementary with angle 6 (it is a supplement to angle 6), then it is 180 - 107, which is 73.1463

You can look at it in many different ways.1475

That is it: see how all of the angle measures are either 73 or 107.1482

Since all of these lines are parallel--these pairs are parallel, and those two pairs of lines are parallel--1488

they are going to have only two different numbers, because all of their relationships are congruent or supplementary.1498

So, it is either going to be congruent, or it is just going to be a supplement to it.1507

Another example: BC is parallel to DE (that is already shown); the measure of angle 1 is 61 (this is 61);1514

the measure of angle 2 is 43; and the measure of angle 3 is 35.1527

This one is going to be a little bit more difficult, because we have lines that are closing in on the sides.1539

And sometimes it is going to be a little confusing, or a little bit hard to see the lines that you need to see.1549

And you are going to have to ignore these.1561

So, look at angle...let's see...3 and angle 4; if you look at BE as a transversal, and these two1564

as the lines that the transversal is intersecting, 3 and 4 are alternate interior angles.1584

But these two lines are not parallel--those two lines that the transversal is intersecting are not parallel.1594

So, you can't assume that they are congruent; you can't say that they are congruent, because look: the two lines are intersecting.1602

Even though they are alternate interior angles, you can't apply the theorem saying that they are congruent, because the lines are not parallel.1612

You have to be very careful; you can't say that angle 4 is 35 degrees.1620

OK, so what can we say? We know that this line segment right here is parallel to this line right here.1626

I can say that the measure of angle 5...because look at this: this angle 5 and angle 2 are alternate interior angles;1641

now, let's see if we can apply the theorem and say that they are congruent.1658

Here is my transversal; here are the two lines that the transversal is intersecting; are the two lines parallel?1663

Yes, they are parallel; now, ignore this side and this side, AD and AE, because you don't need them.1671

It is as if they are not even there; cover it up.1681

Angle 5 doesn't involve those lines; angle 2 doesn't involve those lines.1684

So, all you have to see is this right here; here is BC; there is angle 5 and angle 2.1690

Here, these are parallel; here is 5, and here is 2.1700

So then, these are alternate interior angles, and they are congruent, because their lines are parallel.1708

The measure of angle 5 would be 43.1716

And then, from here, I can say that the measure of angle 7...if you look at angle 7 and angle 1, I have a transversal;1721

there is my angle 7, and there is my angle 1; these two lines are parallel.1752

See how it is only involving the three lines, this line, this line, and this line.1760

Ignore BE; see how I didn't draw it, because it is not involved.1766

Ignore all of the other lines; just look at those three lines for angle 1 and angle 7.1769

If it helps, you can draw it again; this one is a little bit hard to see using this diagram,1774

so if it helps you like this, then just draw it again, just using those three lines.1779

Angle 7 and angle 1 are corresponding; and since the lines are parallel, I can use the postulate to say that angle 1 and angle 7 are congruent.1785

So, the measure of angle 7 is 61.1794

So then, the ones that I found: this is 43; this is 61.1799

OK, to find the measure of angle 4, I can say that, because all these three angles right here form a linear pair,1807

that the measure of angle 7 plus the measure of angle 5 plus the measure of angle 4--they are all going to add up to 180,1824

because they form a straight line; all three angles right here are going to form a 180-degree angle.1833

You can say that the measure of angle...not 1....4, plus 61, plus 43, equals 180.1844

The measure of angle 4, plus 104, equals 180; you subtract the 104, so the measure of angle 4 equals 76; here is 76.1861

All of this is the one that I found, so I will write this in red: 76.1890

And then, let's look at some other ones: now, if you look at angle 8, angle 8 also involves this parallel line.1895

But this one is a little bit harder to see, because you have angle 8 like that; what is this angle right here?1915

This is angles 2 and 3 together; it is this angle and this whole thing.1930

Now, ignore this line; you are just involving this line, this transversal, and this bottom line DE.1936

So, this BE is not there; so it would just be this whole angle together.1946

So then, see how this angle right here and this angle right here are corresponding.1954

But this has another line coming out of it like this to separate it into angles 2 and 3.1961

All I have to do is add up angles 2 and 3, and that is going to be my angle 8.1970

This is going to be 78 degrees, and since the lines are parallel, the corresponding postulate says that they are congruent; that equals 78.1975

I will write that here: 78; and then, this 78 and angle 6 are going to form a linear pair.1993

Right here, 78 +...now, since this is angle 6 right here, you can look at angle 6 and this angle right here,2013

78 degrees, because that is angles 2 and 3 combined; they are going to be consecutive interior angles.2029

And they are supplementary; so you can just do angle 6 + 78 = 180, which is the same thing as looking at this.2038

These are supplementary, so angle 6 and angle 8 (78) are going to add up to 180.2049

It is the same thing: the measure of angle 6, plus 78, is going to equal 180.2055

The measure of angle 6: if I subtract 78, then you get 102, so this is 102.2070

Now, with angle 9, to find the measure of angle 9, that is actually going to involve using the triangles,2088

because the only relationship that this angle has with any of the other angles is that it forms within the triangle.2099

And see how angle 9 is not supplementary; it doesn't form a linear pair; there is no transversal involved with angle 9.2111

It is just these two angles, or those two right there.2120

Angle 9 is actually going to involve what is called the triangle sum theorem, where the three angles of a triangle are going to add up to 180.2125

So, we haven't gone over that yet; if you want, you can just say that the measure of angle 9, plus 61 (this angle),2134

plus the 78, is going to equal 180, and then find the measure of angle 9 that way.2146

You can also look at this big triangle and say that this angle, plus this angle, plus this angle, are going to add up to 180.2151

You can also look at it as this triangle right here, saying the measure of angle 9 plus 75, together, and then 3, are going to be 180.2163

And then, find the measure of angle 9 that way.2175

So, for now, we are just going to solve for these; and that is it for this problem.2179

The last example: Find the values of x, y, and z.2186

Here you have three lines: now, these three lines are going to be parallel.2192

I am going to make them parallel, so that I can solve for these values.2199

Now, the only angle that is given is right here, 118.2203

If you look at this, again, we have four lines involved; and to form special angle relationships, you only need three lines.2209

You need the transversal and the two lines that it intersects to form those pairs of angles.2219

Whichever lines you are using, always keep them in mind, and then look at what line you are not going to use,2228

and ignore that line, since we have four and we only need three.2238

Using this angle right here, 118, I can say that now this one right here and 11z + 8 are corresponding.2245

And then, this one right here and this one right here are alternate exterior angles, because it is involving these three lines, and not this one right here.2262

These would be alternate exterior angles.2273

Or, if I ignore this middle line, and I just say that this transversal with this line and this line2276

(again, ignoring the middle line--pretending it is not there), then 118, this angle right here, with x, would be alternate exterior angles.2290

Imagine if you have a line, a line...here is your transversal; the middle line is not there; this is x, and then this is 118.2303

You see that it is alternate exterior angles.2314

So then, I can say that x is equal to 118, because the lines are parallel.2319

And so then, I can apply the alternate exterior angles theorem, saying that that relationship, that pair, is congruent.2325

The next one: let's look at z; this one right here, 11z + 8, is going to equal 118.2337

Why?--because, if I look at this line, with this line and this transversal, they are going to be corresponding angles.2346

And then, since the lines are parallel, the corresponding angles postulate says that they are congruent.2358

11z + 8 = 118; so if you subtract the 8, 11z = 110; z = 10.2366

There is my x; there is my z; and then, I have to find y now.2387

For my y, I can say that this angle with 118--they are not congruent, remember, because they are going to form a linear pair.2392

They form a line, so they are going to be supplementary.2407

You can also note that this angle is 118, remember, because we said that they were corresponding--this one with this one.2412

So, since this is 118, this angle with this angle would be consecutive interior angles.2421

And if the lines are parallel, then the theorem says that they are supplementary, not congruent.2430

So, either way, 3y + 2 =...not 180; you have to say that this whole thing, plus the 118, is going to equal 180.2436

3y + 2 = 62, and then, if you subtract the 2, then 3y is going to equal 60; y is going to equal 20.2456

x is 118; y is 20; and z is 10; just remember to keep looking for those relationships between the pairs.2474

You can also definitely use the linear pair, if they are supplementary; you can definitely use that.2482

If they are vertical, definitely use that, because you know that vertical angles are congruent.2490

So, any of those things--you have a lot of different concepts that you learn that will help you solve these types of problems.2499

That is it for this lesson; thank you for watching Educator.com.2510