Geometry Trapezoids and Kites

Welcome back to Educator.com.0000

For this lesson, we are going to go over two more types of quadrilaterals, which are trapezoids and kites.0003

First, a trapezoid is a quadrilateral with exactly one pair of parallel sides.0011

We know that this is not a special type of parallelogram, because parallelograms have two pairs of parallel sides.0021

A quadrilateral with one pair--that is the key word; if it only has one pair, it is a trapezoid; if it has two pairs, it is a parallelogram.0028

A trapezoid, here...now, that is the only requirement for a quadrilateral to be considered a trapezoid.0040

It is just that all it has to have is one pair of parallel sides; that is it.0049

Now, this right here...each side of a trapezoid has a special name: here, this is called the base;0054

this is also called the base; base doesn't mean the side on the bottom--the bases are the two sides that are parallel,0067

because we can have a trapezoid that looks like this.0082

That would be considered a trapezoid, because these two are parallel.0087

It does not mean that this is the base; this is not the base; this is the base, and this is the base.0093

It is just that the parallel sides would make it the base.0102

And then, the two other sides are called legs: base, base, leg, leg.0112

And then, these right here would be considered the base angles--these two--those are all base angles.0122

Now, an isosceles trapezoid is a special type of trapezoid.0133

An isosceles trapezoid is when the trapezoid has congruent legs (I am going to erase this, because it is just kind of in the way).0138

We are actually going to go over the angles next.0153

So for now, an isosceles trapezoid is just a trapezoid with congruent legs; when these are congruent, then it is an isosceles trapezoid.0156

If they are not congruent, it is just a trapezoid; they are still trapezoids, but again, if the legs are congruent, then it is an isosceles trapezoid.0166

Here is the isosceles trapezoid; now, we know what an isosceles trapezoid is--a trapezoid,0180

meaning a quadrilateral with one pair of parallel sides, with congruent legs (and these are the other legs).0187

These are the two legs that have to be congruent.0193

Now, for an isosceles trapezoid, both pairs of base angles are congruent.0196

Now, remember how this is the base, and this is the base; so this and this are called the base angles.0204

Now, this would be a pair; this one and this one would be considered base angles; that is one pair.0216

Then, these two angles are considered the second pair of base angles that are congruent.0227

Only for an isosceles trapezoid are base angles congruent.0240

The next one: the diagonals of an isosceles trapezoid are congruent; so all of this is congruent to all of that; diagonals are congruent.0246

The next part of the trapezoid is the median: the median is this middle segment here.0269

But it is not just any middle segment; this point right here has to be the midpoint of this side, of the leg;0279

and this point has to be the midpoint of this leg, and that would make it the median.0290

It is a segment whose endpoints are the midpoints of the legs: leg, leg, midpoint, midpoint,0301

and then you draw the segment that connects those midpoints together, and that would be called the median.0314

Again, to go over isosceles trapezoids, just to review isosceles trapezoids: we know that the legs are congruent;0328

we know that two pairs of base angles are congruent--not opposite pairs of base angles;0338

the top two or the two angles that have to do with one base, and then the other two that have to do with the other base.0349

Those two pairs are congruent, and then the diagonals are congruent (of an isosceles trapezoid).0357

The median is the midpoint of one leg, connected to the midpoint of another leg.0363

And the median can be for any trapezoid; it doesn't just have to be for an isosceles trapezoid.0374

You can have a trapezoid that looks like that; remember: as long as those are parallel, then it is a trapezoid.0378

As long as I have a middle point of that side and the midpoint of this side, and I connect these two, this would be the median.0388

How do you find the median? Let's just sketch this out: the midpoint of this side...the median...and then these are parallel.0403

The median of a trapezoid is parallel to the bases; this median is going to be parallel to the bases.0420

These are the bases, because those two are the parallel sides.0430

The median is parallel, and its measure (this is the median, EF) is one-half the sum of the measures of the bases,0433

meaning that all of this is base 1 plus base 2, divided by 2.0447

So, you are just adding up this base, adding up the other base, and then just dividing that number by 2.0457

Now, that might sound familiar to you; if you add up the two, and you divide by 2, that is also the average.0463

That is how you find the average; so if you are finding the average of the two bases, that would be the measure of the median.0471

So, if you forget this (1/2 times the sum of base 1 and base 2), then you can just remember that as the average of the two bases,0479

because average is pretty easy; you know that you have to add them all up and then divide by however many you have.0492

In this case, you have two bases; so you add up the two bases and divide it by 2, and that is just the median.0499

Now, the kite: most classrooms don't really go over the kite.0510

Most classes, most teachers, skip over kites, because we don't really care too much about kites.0522

But I am going to go over it, just briefly, and just explain to you what a kite is.0531

It is a quadrilateral (we know, because it has four sides) with two pairs of adjacent congruent sides.0537

It is not like a parallelogram, where it has two pairs of opposite sides being congruent.0545

There are still two pairs of congruent sides, but then, they are adjacent.0554

That means that this side and this side are congruent, and then these other remaining two sides are congruent; that is a kite.0563

Also, the diagonals are perpendicular; and then, the long diagonal, which is this right here, bisects the angles.0573

It is not both diagonals; it is not like the rhombus, where the diagonals bisect all of the angles.0610

It is just this right here--it is just these angles that are bisected.0619

We know that the two pairs of adjacent sides are congruent; the diagonals are perpendicular;0626

and then, only those are congruent, and then those are congruent.0634

They are not congruent to each other; just this angle is bisected, and then this angle is bisected.0641

And that is pretty much it for kites: again, one more time, a kite is a quadrilateral with two pairs of adjacent congruent sides.0649

So, these two are congruent, and then the other two are congruent to each other--not the opposites.0658

Diagonals are perpendicular, and then, only this angle and this angle are bisected by the diagonal.0664

The longer diagonal bisects each of the angles.0673

And that is the kite.0677

Let's fill in these lines, going over the different types of quadrilaterals that we went over.0683

This top part, right here, is going to be a quadrilateral; so then, this whole thing is on quadrilaterals.0693

Then, we went over three different types of quadrilaterals.0714

The first one, the one with all of this stuff below it, is parallelograms.0717

How do I know that this one is parallelograms?--because I have a lot of things that I can write under here--the different types of parallelograms.0728

What are they? The two different types of parallelograms that we went over are rectangle and rhombus.0736

Then, what about this right here? This is the square.0748

What is this right here? This is another type of quadrilateral that is not a parallelogram, so it doesn't go below the parallelogram; it is the trapezoid.0757

And then, we went over a special type of trapezoid: if the legs are congruent, then it would be an isosceles trapezoid.0773

And then, a third type of quadrilateral that we went over briefly is the kite.0787

And there are no special types of kite, so that is just kite; that is it.0794

This will help you for those questions with "always/sometimes/never."0801

We know that a quadrilateral is sometimes anything below it; when it goes down, it is "sometimes."0808

A quadrilateral is [always/sometimes/never] an isosceles trapezoid--well, it is "sometimes,"0815

because sometimes it is a trapezoid, and sometimes that is an isosceles trapezoid.0820

A parallelogram is sometimes a square.0825

A rhombus is always a parallelogram; a square is always a quadrilateral.0833

Remember: if you are going upwards on this chart, then it is "always."0840

A kite is always a quadrilateral; a kite always has four sides.0844

Now, when you go side-to-side, that is when it is "never," because a parallelogram is never going to be a trapezoid.0849

It is either going to have one pair of parallel sides or two pairs of parallel sides, to make it one or the other.0857

So, when it goes side-by-side, then it is "never"; a parallelogram is never going to be a trapezoid.0865

Or a rectangle is never going to be an isosceles trapezoid, or a square is never going to be a kite.0871

Anything that is side-by-side, that is not going up the arrows, is going to be "never"; this will help you with that.0880

Let's go over examples: State whether each statement is true or false, based on isosceles trapezoid ABCD.0891

Let me just label this out: here is A, B, C, and D; I will label this as E.0900

Isosceles trapezoid: that means that we know that these are parallel, and these are congruent.0912

We also know that this diagonal is congruent to this diagonal, and we know0922

that this pair of base angles is congruent, and then the top pair of base angles is congruent.0933

The first one: AC is congruent to BD--now, remember: that is only true for isosceles trapezoids.0943

If it is a regular trapezoid, then none of these properties apply to them; it is just isosceles trapezoids.0956

Irregular trapezoids are not isosceles trapezoids; the only thing they have is the pair of parallel sides--that is it.0964

All of these extra properties are all only for isosceles trapezoids.0975

So, AC is congruent to BD, because it is an isosceles trapezoid; this one is true...and this needs that, and so does this.0980

The next one: AC is perpendicular to BD.0993

Now, there was no property or theorem that went over that.1003

It could be, but there is nothing that we went over that says that AC is perpendicular to BD.1005

The only thing that we went over off our diagram is that they are congruent.1017

So, in this case, this is false; it could be true, but we don't want to say that it is true when it could be false, so we are just going to say that it is false.1023

The next one: Angle ABC and angle BCD are supplementary.1035

Now, think about it: here is AB, extended; here is BC, extended; remember: these are parallel.1051

Then, let's say that BC is a transversal; this is angle ABC, so this angle...and then angle BCD, this angle--are they supplementary?1063

Yes, they are, because they are consecutive interior angles (or you can say that they are same-side interior angles).1080

When it comes to consecutive interior angles, we know that they are supplementary, as long as these lines are parallel (which we know they are).1091

These two angles are supplementary; they are going to add up to 180; this one is true.1098

The next example: EF is the median of trapezoid ABCD.1108

If it is a median, we know that these are congruent, and these are congruent.1114

Now, the reason why I am not marking these parts congruent to these parts is because I don't know that it is an isosceles trapezoid.1123

This leg can be different than this leg; so then, I just have to mark this as the median of this whole leg, and then this as the median of this leg, separately.1134

So, I know that AD is 5; this is 5; BC is 13.1148

Remember: to find the median EF, EF is (let's just write it here) one-half (AD + BC)--1154

in other words, AD + BC, divided by 2, or the average of the bases, which are these two.1175

AD is 5 (let me write...); EF equals 1/2(5 + 13); so then, this right here is 18, divided by 2 is 9; so EF = 9.1189

The next one: BC is 15; EF is 11; find AD.1215

Now, they give you the median, and they give you BC, one of the bases; but they are asking for the second base.1226

I am just going to use the same formula right here, and I am going to plug in 11 for EF; that equals 1/2...1234

what do they give me?...BC, so then...AD...I am going to write it as my variable, plus 15.1245

With this, I can either distribute the 1/2, or I can say 11 = (AD + 15)/2.1260

And that way, it will be easier, because I can just multiply the 2 over to both sides, and I get 22 = AD + 15.1271

Subtract the 15, and I get 7; so then, AD is equal to 7.1280

So then, if they give you a median, and they are asking for one of the bases, then you just leave that as your variable.1291

The next one: AD is x + 4; EF is 12; and BC--this is BC--is 2x + 2; find x...it is the same thing here.1303

EF is 12; that equals...and I am just going to put it over 2, so then AD, which is x + 4, plus BC, 2x + 2, all over 2.1319

So then, this becomes 3x + 6, divided by 2; again, multiply this by 2; 24 = 3x + 6; subtract the 6; I get 18 = 3x, and x = 9.1339

And then, if you want to double-check your answer, you can just plug it back in.1372

So, 9 + 4 is 13...and then, this is 18; this is 20; 13 plus...1375

Oh, let's look at this again: we have x + 4 + 2x + 2; 3x + 6, divided by 2...1395

we multiply the 2 over, so we have 24 = 3x...OK, I divided it wrong, so it is a good thing that I double-checked.1412

This is...x is 6; that means that these would be different.1424

It is always good to double-check your answers: 6; this is 10; and this is 12, plus 2 is 14.1434

So then, that adds up to 24, divided by 2 is 12; so then, this is the correct answer.1444

OK, the next one: Example #3: Complete each statement with "always," "sometimes," or "never."1452

Now, this had to do with that flowchart that we did, where, if you are going down on the chart, then it is going to be "sometimes";1462

if you are going up on the chart, it is going to be "always"; and if you go side-by-side on the chart, then it is going to be "never."1470

So, if you need to write it out, then go ahead; if not, then let's just try to do it without it.1475

If you want to draw it out, then that is up to you.1486

A square is [always/sometimes/never] a rectangle--again, a square is a type of rectangle, just like a Chihuahua is a type of dog.1491

So, a Chihuahua is always a dog, so a square is always a rectangle.1502

A quadrilateral is [always/sometimes/never] a rhombus.1513

Well, a quadrilateral could be a parallelogram; it could be a rectangle; it could be a trapezoid.1518

A quadrilateral can be a lot of different things, so it is only sometimes a rhombus.1526

A trapezoid is [always/sometimes/never] a parallelogram.1537

This one, we know, is "never," because in order for it to be considered a parallelogram,1541

it has to have two pairs of parallel sides; trapezoids only have one pair.1547

A parallelogram is [always/sometimes/never] a square.1554

Well, a parallelogram could be a rectangle, or it could be a rhombus; a rectangle is not always a square--1560

a rectangle can just remain a rectangle; a rhombus can remain a rhombus.1574

So, a parallelogram is not always going to be a square--it is "sometimes."1582

And it is also going down on the chart, so it is "sometimes."1586

A rhombus is [always/sometimes/never] a trapezoid.1592

A rhombus is a type of parallelogram; we know that parallelograms and trapezoids are two different things, so it is "never."1599

And the last one: Determine if the figure is a trapezoid.1610

Now, first, we have to think, "OK, what makes a trapezoid?"1615

The only property of a trapezoid is the one pair of parallel sides--only one--that is it.1620

All of the other properties, with the legs being congruent, diagonals being congruent, two pairs of base angles being congruent--1632

those properties have to do with isosceles trapezoids.1641

This one just says "trapezoid"; we are only talking about a trapezoid.1645

The only property that would make it a trapezoid is that one pair of parallel sides--that is all we have to do--we don't have to do anything else.1651

So, here, we know that, if we were to have one pair of parallel sides--if it is a trapezoid--1662

then it has to be these two sides here, because we know that this and this are not going to be parallel.1670

So, the slope of this is going to be, remember, rise over run; we are going to count...1678

Rise is going up and down; run is going side by side.1689

If you go up, remember, it is a positive number; if you go down, it is a negative number.1693

If you go to the right, it is a positive number; and if you move to the left, it is a negative number.1699

Here, let's go up; we are going to find the slope of this right here.1706

And again, we are finding the slope of this one and this one, because, if they are parallel, then they will have the same slope.1711

We just have to find the slopes; and if they are the same, then those two will be parallel.1719

So then, the slope of this is going to be 1, 2; so that is +2, over 1, 2, 3, 4.1726

It is going to the right, so that is positive, so it is positive 4; that means that the slope of this is 1/2.1737

Now, if you are still a little unfamiliar with slope, I can also go down, just to show you.1746

If I go down 2, that is -2, over...and then I am going to go to the left 1, 2, 3, 4; that is -4, because I went to the left.1755

This also becomes 1/2, so it is the same; I just wanted to show that, if you go from this point to this point,1771

you are going to get the same slope as when you go from this point to that point.1779

Let's look for the slope of this one right here: from here to here, I get...actually, let's go from here to here.1785

I am going to go down 3: 1, 2, and then one more--3: that is -3.1795

And then, I have to go to the left 1, 2, 3, 4, 5, 6; that is -6, which is 1/2.1805

So, this slope is 1/2, and this slope is 1/2; therefore, this is a trapezoid--"yes"--because these two are now parallel.1816

So yes, it is a trapezoid.1834

That is it for this lesson; thank you for watching Educator.com.1843