Geometry Parallelograms

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The next lesson is on parallelograms; and before we talk about parallelograms, let's talk about quadrilaterals.0002

A quadrilateral is a four-sided polygon; now, we know that a polygon is a shape that is closed off.0009

It is a shape with three or more sides that is completely closed, and each side has to be a straight line.0020

When we have a polygon that is four-sided, that is a quadrilateral.0028

This is considered a quadrilateral; this is a quadrilateral; any type of shape where I have four sides is a quadrilateral.0033

Now, some non-examples, some examples that are not quadrilaterals, would be maybe something like that,0047

where it doesn't close; that is not a quadrilateral; or if I have something that overlaps like that,0056

if two sides overlap, this is not a quadrilateral; and if I have something that maybe is curved like that,0069

a side that is curved, that would not be a quadrilateral, as well.0079

Again, a quadrilateral is a four-sided polygon, something that is four-sided.0083

Each side has to be a straight line, and it is closed, and there are no overlapping sides.0089

A parallelogram is a special type of quadrilateral, meaning that it is a four-sided polygon, but it has two pairs of parallel sides.0098

That is why it is called a parallelogram, because there are two pairs of parallel sides.0108

To show that my sides are parallel, I can do this; that means that those sides are parallel.0117

To show that these sides are parallel, I am going to draw two of them.0126

These two are parallel, and then these two are parallel; so we have two pairs of parallel sides.0136

Now, if I label this as A, B, C, and D, then I can name this parallelogram by its symbol; that would be the symbol for a parallelogram.0142

Parallelogram ABCD: that is how I can name it, just like when I have a triangle; I can say triangle ABC.0159

In the same way, I can say that this is the parallelogram ABCD.0169

I can also say parallelogram BCDA; I could say parallelogram CDAB, and so on, as long as the four vertices that I name are in order.0173

So, I can say ADCB; I can say DABC; but I can't say parallelogram ACBD, because it skips.0192

You are skipping a vertex; when you label your parallelogram, make sure that they are in order--you are calling it in order.0210

It could be this way; it could be to the right, clockwise, or counterclockwise; it does not matter, as long as they are back-to-back.0222

This is something that you cannot say.0230

Now, when we talk about opposite pairs, we have two pairs of opposite sides and two pairs of opposite angles.0233

I know that side AB and side DC are opposite sides, and angle A with angle C are opposite angles.0243

And angle B with angle D are opposite angles.0258

Some properties of parallelograms: now, these properties are in the form of theorems,0270

just so you know that these are considered theorems.0278

The first property is that opposite sides of a parallelogram are congruent.0282

As long as we have a parallelogram, and you know that it is a parallelogram, they either tell you that it is a parallelogram,0287

or if they show you these symbols; the definition of a parallelogram is a quadrilateral with two pairs of opposite sides that are parallel.0294

That is the definition of a parallelogram; that is not a property--that is what it means to be a parallelogram, the definition.0307

The properties are these, saying that, if this is a parallelogram, then opposite sides are congruent.0315

That means that this side with this opposite side are congruent, and this side with this side are congruent.0324

So then, if this is 5, then this will also be 5; if this is 8, then this is also 8.0333

Opposite sides are congruent; there will be two pairs--there is one pair, and this is the other pair.0341

Opposite angles of a parallelogram are congruent; that is the next property.0351

That means that this angle and this angle are congruent, and that angle with this angle are congruent.0357

So, as long as we have a parallelogram like that, then I know that opposite angles are congruent.0365

So far, opposite sides are congruent, and opposite angles are congruent.0378

Then, the next one: Consecutive angles in a parallelogram are supplementary.0386

So then, I am just going to show that...I don't have to do this, because I am already telling you that this is a parallelogram;0398

but just so you get used to seeing that opposite sides are parallel...0404

Consecutive angles of a parallelogram: that means this angle, so if I have angle 1 and 2, they are going to be supplementary.0409

And one way I can show you why this is true, or how this is true: if I extend this side out, and extend this side out,0420

then extend this side, then...now, I know that these two sides are parallel; this is like a transversal.0430

If you remember that one section, that chapter that we went over, about parallel lines being cut by a transversal,0442

then remember that we had special angle relationships; so remember how this angle right here,0450

and (I am actually going to call that angle 2, just so that it is the same as this diagram right here)0456

this is angle 1, and this is angle 2; now, we know that the name0460

for this angle relationship is consecutive interior angles, or same-side interior angles.0466

And we know that they are supplementary; as long as the lines are parallel, then these angles are supplementary.0474

So, it is the same thing here: these are consecutive interior angles.0479

Then these are supplementary; they are all the consecutive angles in a parallelogram.0486

So, if I know that these two lines are parallel, this is a transversal;0493

then, my interior angles are supplementary, in the same way that consecutive interior angles are supplementary.0497

Then, these two are supplementary; this angle with this angle is supplementary;0507

this angle with this angle is supplementary; and this angle with this angle is supplementary.0513

Opposite angles are congruent, but consecutive angles are supplementary.0517

Diagonals of a parallelogram bisect each other.0524

If I draw diagonals, those are my diagonals; now, they are not congruent--diagonals are not congruent.0528

That is a common mistake made by students--saying that these diagonals are congruent.0541

Diagonals of a parallelogram are not congruent; obviously, it looks like this...from here...0549

if you were to walk from this point to this point, that looks further than this point to this point.0554

So, you can't say that this diagonal is the same as this diagonal.0563

Diagonals are not congruent; they actually just bisect each other--that means that this diagonal cuts this diagonal in half;0572

this got cut in half; and then, this diagonal cuts this diagonal in half, so that means these are congruent.0582

The two diagonals are not congruent to each other; but the two halves of each diagonal are congruent.0591

So then, only this part and this part of the same diagonal are congruent; this part and this part of the same diagonal are congruent.0601

Another mistake is saying that this part, this half, of this diagonal is congruent to this half of the other diagonal.0608

No; only the diagonal is cut in half--each diagonal is just cut in half; that is all you have to think of.0615

They just bisect each other, and that is it.0621

We went over four properties so far, before we start our next example.0625

The first property of parallelograms...0631

Well, first of all, the definition of parallelogram says that we have a quadrilateral (meaning that it is a four-sided polygon)0633

with two pairs of opposite sides parallel; that is a parallelogram.0642

The properties: there are four of them--the first one is that opposite sides are congruent;0649

opposite angles are congruent; consecutive angles of a parallelogram are supplementary;0657

and the last one is that diagonals bisect each other--those are four properties.0664

So, using those properties, let's continue with our examples.0671

The first one: complete each statement about the parallelogram.0677

I know that I have a parallelogram; BC (there is that side) is congruent to what?0681

You are going to complete the statement: BC is congruent to what?--the side opposite, so AD.0688

Now, I know that this sounds really simple; it looks like the only other side that could be congruent to BC is AD.0697

But sometimes parallelograms are not so stretched out like that; sometimes this side and this side could look a lot closer in length.0704

Just don't assume; if it is just a parallelogram, even if side BC and this side right here, CD, look like they are congruent, you can't assume that.0720

So, all we can assume from parallelograms when it comes to their sides is that opposite sides are congruent.0733

AE is congruent to BE? No, AE is congruent to DE? No, AE is congruent to CE.0742

This diagonal is just cut in half; so AE is congruent to CE, the other half of the same diagonal.0753

AD is parallel to what? AD is parallel to BC.0763

And then, angle BCD is congruent to the angle opposite.0770

Now, we are just talking about the whole angle, not just these parts.0777

So then, the whole angle, BCD, is congruent to angle D (since B and D are corresponding)...DAB.0782

That is Example 1; the next example is to find the values of x, y, and z of the parallelogram.0798

Here is x; there is y; and there is z.0807

Since I know that opposite sides are parallel and congruent (these sides are congruent;0812

these sides are congruent), I can say that z...let's just say x =, y =, and z =; and that way,0822

I have all of my answers right here...so I know that z = 11.0835

Now, for x, if you look here, remember: we know that opposite sides are parallel.0840

So, if I just extend these out, see how these are my parallel lines, and this is my transversal.0848

If I were to draw that out again, there is BC; this is AD; and then, here is my transversal, CA; this is C, and this is A.0860

Now, if this is parallel, and this is 40, and this is x, what can I say about x?0874

If lines are parallel, then alternate interior angles are congruent.0884

So, if this is 40, this and this are alternate interior angles, so x has to be 40 degrees.0892

And then, for y...well, I know that y and this one are congruent, and angle B, but they don't give me angle B, so I don't know--0903

I can't say that y is the same as this angle, because I don't have that angle.0915

I have to find y in another way, so let's see...I know that x is 40; this is 40; what is this angle all together?0920

This is 65 + 40; this is 105 degrees; I know that consecutive angles are supplementary, so if this is 105, then this has to be the supplement to 105.0934

So, the measure of angle y equals 180 - 105; or you can say 105 + the measure of angle y equals 180; you can do whichever.0954

Here, this is 75; so the measure of angle y is 75 degrees.0971

Again, all I did here to find y is find the whole angle right here, because that is what makes up this angle of this parallelogram;0980

and then this angle with this angle is supplementary.0989

The next one: Find the distance of each side to verify the parallelogram.0997

I need to find the distance of AB and the distance of BC, this distance and this distance,1003

and then compare them, because I know that opposite sides are congruent.1008

So then, as long as the distance of A to D, this right here, and this, are congruent,1013

and then this and this are congruent, then that is verifying that it is a parallelogram.1020

Now, they are just telling you that it is a parallelogram; you are just verifying.1024

You are just using the distance formula to just show...you are just writing down...1028

now, you know that it is going to be the same, because it is a parallelogram.1036

So, they are just asking you to verify that.1039

The reason why I am saying this is because, in the next lesson, we are actually going to prove parallelograms.1043

We are going to use the distance formula and other theorems and such to actually prove that a quadrilateral is a parallelogram.1047

But this is not proving; you are not proving that it is a parallelogram--you are just verifying,1057

meaning that it is a parallelogram, but you are just showing it; just actually write the numbers to show that it is a parallelogram.1061

The distance formula, we know, is the square root of (x₁ - x₂)...and this doesn't mean x squared,1068

or anything like that; it is just the first x, minus the second x, squared; plus y₁, the first y, minus the second y, squared.1082

So then, let's find the distance of AB; AB is going to be the square root of...1094

Oh, before we begin, let's actually write out the points of what these are.1104

B is 1, 2, 3; that is (-3,2); C is 1, comma, 1, 2, 3; D is at 1, 2, 3, comma, -1, -2; and A is at -1, -1, 2, 3.1109

That way, I have all of my points.1140

So, if I am trying to find the distance of A to B, then I am going to have to use these points.1142

x₁ (it doesn't matter which one you use) is -1, minus -3, squared, plus y₁, the first y, -3, minus 2, squared.1147

-1 minus -3; now, if you remember from algebra, minus a negative makes a plus.1169

If you have two negative signs next to each other like that, that just makes both of them into a plus.1179

-1 + 3 is 2; that is 2 squared; plus...-3 - 2; that is -5, squared; this is 4 plus 25; and this is the square root of 29.1185

And this we can just leave like that, unless your teacher wants you to round it to the nearest hundredth, or some decimal number.1214

Then you would have to change this to a decimal.1225

Otherwise, you can just leave it like that; I am pretty sure that you can probably just leave it like that.1230

Then, I am going to find the side opposite to show that they are congruent: CD.1235

CD is, let's see...we can use whichever number, so let's do 3, the first x, minus...we will call that the second x, squared,1243

plus the first y, minus the second y, squared; so here, this is 3 - 1, is 2; that is 2 squared, plus -2 - 3 is -5, squared;1255

that is 4 + 25, which is the square root of 29.1275

Since these are congruent, we know that these sides are congruent.1287

Let's do BC: BC = √[(-3 [the first x] - 1)² + (2 - 3)²].1292

The square root of...-3 - 1 is -4, squared, plus...this is -1, squared; equals...16 + 1, which equals the square root of 17.1309

Then, AD, the side opposite that, is going to be -1 - 3, squared, plus -3 - -2, squared;1330

the square root of...-1 - 3 is -4, squared; plus...minus a negative...they both make a plus, so this is -1, squared;1350

4 times 4 is 16, plus 1, which is the square root of 17.1366

Now, see how this is equal to this, and this is equal to this.1374

So then, I know that AB is congruent to CD, and BC is congruent to AD.1384

OK, for the next example, we are going to use the same diagram, the same parallelogram.1396

And we are just going to verify that it is a parallelogram by using slope.1404

Now, how would we use slope to verify the parallelogram?1409

If I find the slope of BC and the slope of AD, they are parallel; we know that they are parallel, because it is a parallelogram.1413

So, if we find the slope of this, and we find the slope of this, they should be the same,1424

because we know that two lines, when they are parallel, have the same slope.1429

The same thing works here: we are going to find the slope of this, and find the slope of this, to verify that it is a parallelogram.1436

Now, if you want, you can find the points of each one of these, like we did in the previous example.1448

Find the coordinates of each; and you can use the slope formula.1455

The slope formula is y₂ - y₁, meaning the difference of the y's, over the difference of the x's.1460

You could do it that way, using the point of this and using the point of that, and then solve it that way.1471

Or, since we actually have a coordinate plane, and we have all of the points there, we can just do rise over run,1477

meaning we can just count how many we are rising and how many we are running.1488

So then, the slope of BC: to go from this point to this point, you are going to rise one,1492

meaning you are going to go up one (and rise just means how many you are going to go up and down).1499

Run is how many you are going to go left and right.1504

We are going up 1; since we are going up 1, that is a positive 1, over...1, 2, 3, 4; and that is going to the right, so we know that it is a positive number.1507

Whenever you go up, it is a positive, because remember: the y-axis...see how when you go up, you are going positive.1519

But when you go down, you are going negative; see how these numbers are all negative.1527

For the x-axis, the same thing happens: when you go right, you are going towards positive numbers;1531

when you go to the left, you are going towards the negative numbers.1536

So, if you go up, you are going positive; if you go right, you go positive; going down is negative; left is negative.1539

So, we went up; that is a positive 1; and 1, 2, 3, 4--you went to the right 4; that is a positive 4.1545

That means that the slope of BC is 1/4.1552

For CD, we are going 2 to the right (that is 2), over...how many are we going down?...1, 2, 3, 4, 5.1558

That is a negative 5, because we went down 5; the slope of CD is -2/5.1572

Now, it doesn't matter if you go from D to C--it will be the same thing.1584

If we count from D up to C, we are going to go up 5 (that is a positive 5)...1589

Oh, I'm sorry; I did it the wrong way.1597

Let's see, let me just try to draw this over again; OK.1607

Let's try that again: I am going from C to D.1611

Now, I did my rise and run; I did it run over rise--now, be careful not to make that same mistake, OK? I'm sorry about that.1616

Now, we are going run...if we go and run 2, I have to write that as my denominator, 2,1622

because that is the run; that is the bottom part; that is the denominator.1630

And then, I am going down 5; so that is a rise.1633

It is actually more helpful if you just do the rise, the up and down, first.1640

So then, I would have to go down first, and then to the right; that would be a lot easier, and I wouldn't have made that mistake.1644

So then, it is better to just go down...rise first (meaning up and down first), and then you go left and right.1650

Here, you can go down 1, 2, 3, 4, 5; that is -5, over 2; that is a positive 2.1657

The same thing happens if I go from D to C; I am going to go up 5 (that is a positive 5), over left 2 (that is a negative 2).1665

It is the same exact thing.1675

Now, the slope of AD will be...let's go from A to D; I am going to go up 1; that is a positive 1, over 1, 2, 3, 4; that is a positive 1/4.1679

See how they are the same.1693

Then, the slope of AB is going to be...I go from A to B.1696

So then, that would be from A..."rise"--going up 1, 2, 3, 4, 5, positive 5, over left 2; that is negative 2; and that is the same as that.1703

So then again, I am going to show that this is parallel to this; they are parallel.1716

And this is the same as this, which means that these are parallel.1723

We are just verifying that they are parallel, and that this is, in fact, a parallelogram.1727

In the next lesson, we are actually going to do pretty much the same thing,1734

except we are going to prove that they are parallelograms.1737

Using the properties, and using the theorems, we are going to actually prove that a quadrilateral is a parallelogram.1742

That is it for this lesson; we will see you soon--thank you for watching Educator.com.1748