Geometry Rectangles

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The next lesson is on rectangles.0001

Now, a rectangle is a quadrilateral with four right angles; we know that there are only four angles in a rectangle, and all four are right angles.0005

Now, rectangles are a special type of parallelogram; if I have a quadrilateral, we know that a quadrilateral is just a polygon0017

with four sides--any polygon with four sides is considered a quadrilateral.0027

Then, a special type of quadrilateral is a parallelogram; and then, a special type of parallelogram is a rectangle.0032

That means that a rectangle has all of the properties of a quadrilateral and a parallelogram.0053

Now, a quadrilateral doesn't really have any properties, except that it just has four sides.0060

For a parallelogram, though, we have a few: just to review: for a parallelogram, we know that opposite sides are parallel.0065

We know that two pairs of opposite sides are congruent; we know that two pairs of opposite angles are congruent.0072

We know that diagonals bisect each other; and we know that consecutive angles are supplementary.0082

Those are all of the properties of a parallelogram.0090

Now, since a rectangle is a special type of parallelogram, all of those properties of parallelograms apply to rectangles,0093

which means that rectangles have opposite sides parallel and congruent (this is congruent also);0101

we know that opposite angles are congruent; so this angle and this angle--in this case, all angles would be congruent0115

to each other, because right angles are all congruent; we know that rectangles' diagonals bisect each other,0122

so we know that these bisect each other; and we are actually going to go over more specifically the diagonals of rectangles in a second.0138

And we know that consecutive angles (the measure of angle B plus the measure of angle C) are 180.0149

We know that that is true, because this is a right angle, so that is 90; a right angle--that is 90; together they make 180, which is supplementary.0156

All of the properties of a parallelogram apply to the rectangle.0166

The diagonals of rectangles: if a parallelogram is a rectangle, then its diagonals are congruent.0174

If we have a rectangle, this diagonal and this diagonal are going to be congruent.0181

That means that the distance from here to here is the same as the distance from here to here.0191

So then, this diagonal and this diagonal are congruent.0200

If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.0211

We know that a parallelogram's diagonals are not congruent; they could be, but they are not always.0217

So, that wouldn't be considered a property of a parallelogram.0224

The property of a parallelogram on diagonals is just that they bisect each other.0227

And these congruent diagonals bisect each other, too; but they bisect each other, and they are congruent.0232

It is like an additional property added on when you have rectangles.0240

It is that all of the properties of a parallelogram are true, plus more.0246

If the diagonals of a parallelogram are congruent, then it is a rectangle.0250

These are the same theorem; it is just that this is the converse, because it is saying that,0260

if it is a rectangle, then we know that the diagonals are congruent.0266

But also, if the diagonals are congruent, then it is a rectangle; so it works vice versa--it works both ways.0269

For this one, we are going to prove that this is a rectangle, or we are going to prove that it is not a rectangle.0282

Determine whether the parallelogram is a rectangle.0291

We know that these are parallel, because the slope of AB and the slope of DC are going to be the same,0295

and the slope of AD and the slope of BC are the same.0304

So, we know that it is a parallelogram; and automatically, we can assume that, even if we don't know what the slopes are,0307

we can assume that these opposite lines are going to have the same slope, because it tells us that it is a parallelogram.0312

So, now, what I want to know is if it is going to be a rectangle.0321

How do we know that this would be a rectangle?0331

We can still use slope; since this is already a parallelogram, we don't have to show their slopes--0334

well, we do, but we don't have to show that opposite slopes are the same,0344

because again, they tell us that it is a parallelogram.0350

But what we do have to do is show that, since we know that slopes that are perpendicular have negative reciprocals,0354

I know that this side, AB, and side BC are going to be perpendicular, because it is right angles.0365

So, since these are right angles, in order for that to be a rectangle, their slopes have to be negative reciprocals of each other.0372

I am going to find their slopes: let's see, the slope of AB is going to be, remember, rise over run.0380

Slope is rise over run, how many you go up and down versus how many you go left and right.0390

Remember: if you count up, that is a positive number; if you count down, that is a negative number.0401

If you count right, that is a positive; and if you count left, then that is a negative.0407

And we know that because, on the y-axis, as you go up, the numbers get bigger; they become more positive.0414

If we go to the right, the same thing happens: the numbers go towards the positive numbers.0421

If you go down, those are the negative numbers--you are going towards negatives.0427

And then, if you go to the left, then the numbers are getting smaller again, to the negatives, so it is a negative number.0431

Let's count up from here to here: to count the slope, you are going to go 1, 2, 3; that is a positive 3; you went up positive 3.0437

And then, we are going to run 1, 2, 3, 4; so the slope of AB is positive 3 over 4.0449

And then, let's see, the slope of BC: again, we have to go up and down first.0459

I am going to go down: 1, 2, 3; I went down 3--that is negative 3, over...0469

1, 2--positive 2; so this one was 3/4, and this one is -3/2.0481

Now, they are not negative reciprocals of each other; this one is positive, and this one is negative,0493

but then the negative reciprocal would be -4/3, but it is not.0500

So, automatically, I don't have to go on anymore; I just know that these two are not perpendicular.0505

Therefore, this is not a right angle, which means that this is not a rectangle.0511

Now, if these two were negative reciprocals of each other, then I would have to continue and find the slope of DC,0521

and then find the slope of AD, and make sure that they are also reciprocals of each other,0529

because this could be a right angle, but then, this might not, or the other ones might not.0535

But in this case, if they tell us that it is a parallelogram, then as long as we have one angle that is a right angle,0543

then it will be the same for all, because it is a parallelogram, so opposite angles would have to be congruent.0552

Let's just do a little summary of rectangles: it is pretty much all of the properties of a parallelogram, plus "all angles are right angles" and "diagonals are congruent."0564

Opposite sides are congruent and parallel; those are both properties of a parallelogram.0583

Opposite angles are congruent; that is also a property of a parallelogram.0592

Consecutive angles are supplementary; that is a property of a parallelogram.0600

Diagonals are congruent and bisect each other: well, for the property of a parallelogram, it is just that they bisect each other.0606

For parallelograms, they are not always going to be congruent; so only these are the ones that are parallelograms:0614

properties of parallelograms--that is the symbol for a parallelogram.0625

Now, then new ones, the ones that are more specific to rectangles: diagonals are congruent--that is this one.0633

And all four angles are right angles--there is another one that is just considered a property of a rectangle.0641

All of the ones in red are properties of rectangles.0651

Let's work on our examples: the first one: we are going to find the value of x.0664

The measure of angle x here is 10x + 5, right here; and the measure of angle 2 is 55 degrees.0669

Now, since these two angles, we know, are what?--they add up to 90--they are complementary, I just have to make those two,0678

10x + 5, plus 55, equal to 90 degrees; so here, this is going to be 10x + 60 = 90.0698

Subtract the 60, so 10x = 30; and x = 3.0716

Oh, and then, just to look back, they were asking us for the value of x.0730

If they asked us to find the actual angle measures, well, we have the measure of angle 2; and then, we would have to take this x,0735

and plug it back into this right here, so that we would find the measure of angle 1.0745

And if we do, 3 times 10 is 30, plus 5 is 35, which would add up to 90; so then, that would be correct.0751

But they are only asking us for the value of x, so that would be the answer.0762

Now, the next one: here is the diagonal; AC is 52, and DB (which is the other diagonal) is 5x + 2.0766

Now, if you remember, the property of a rectangle is that diagonals are congruent.0777

So then, we know that AC is congruent to DB; that means that I can just make them equal to each other.0784

52 = 5x + 2; I am going to subtract the two, so I get 50 = 5x...divide the 5, and so, 10 is x, or x = 10.0798

Both of these use the properties of the rectangle that are more specific to the rectangle,0817

where each angle of a rectangle is a right angle, and the diagonals are congruent.0824

The next example: Name all congruent sides and angles.0833

All we have to do is just name all of the sides that are congruent and all of the angles that are congruent.0838

First, I know that AB is congruent to CD (we are doing sides first), and then AD is congruent to CB.0848

And then, the diagonals (because this is a rectangle): DB is congruent to CA.0871

And now for the angles: this is a little bit different, because...well, we know that all of the angles are congruent.0888

Angle A is congruent to angle B, which is congruent to angle C, and it is also congruent to angle D; why?0898

Because all right angles are congruent; we know that each of these is a right angle.0911

But then, we have these diagonals written here; so that means that all of the angles are split up now.0918

So, we have to see what angles are congruent here.0924

Since we know that these sides are parallel (that is a property of a parallelogram, and all properties of parallelograms apply to rectangles),0929

angle 1 and angle 7 are alternate interior angles; so if I were to draw that again, here is, let's say, AB extended.0942

Here is DC extended; here is a transversal; here is 1; there is 7; so we know that angles 1 and 7 are congruent, because they are alternate interior angles.0951

That is congruent to angle 7; we also know that angle 2, then, is congruent to angle 8, because of the same reason, alternate interior angles.0975

Now, I don't know if you can see this; but if, for this triangle here, since we know that these diagonals bisect each other,0992

all of these are actually equal parts; so if you can see that this is a triangle (an isosceles triangle, to be more specific),1013

because this side and this side are congruent, then we know that these angles are congruent,1029

because of the base angles theorem, or you can say the isosceles triangle theorem.1037

I know that angles 2 and 5 are congruent, because of that reason there.1045

Now, I am just going to add that onto this right here, because angle 2 is congruent to angle 8; but angle 2 is also congruent to angle 5,1052

which means that (I am trying to get that red)...all of those would be congruent.1067

2 is congruent to 8, and 2 is congruent to 5, and 5 is congruent to 4, because they are alternate interior angles; so all four of those angles are congruent.1080

In the same way, 1 is congruent to angle 7, and since these are congruent, then I know1095

that 6 and 7 are congruent, because of the base angles theorem of this triangle.1108

And then, this is alternate interior angles with angle 3, so all four of those angles are congruent.1117

Now, we also have these four angles in the middle right here.1132

Now, I can't say for sure if these angles are going to be congruent to any of the outer angles,1136

because we just can't say; we don't know what the measures of those angles are.1143

But what I can say is that angle 9 is congruent to angle 12, because they are vertical angles.1150

And then, angle 10 is congruent to angle 11, because they are vertical, as well.1164

So, there is a lot of stuff right there: all of the congruent sides and the congruent angles.1172

For the next example, we are going to determine whether each statement is always true, sometimes true, or never true.1181

If a quadrilateral is a rectangle, then it is a parallelogram.1190

Now, if I say "quadrilateral," that is a very general name for a polygon.1199

And then, it gets more specific to a parallelogram, and then it gives another more specific name, a type of parallelogram, to a rectangle.1218

This is showing you where it starts from: a quadrilateral is the big picture; it is what the general polygon is called.1241

Then, it is down to this type; if it is like this, then it is a parallelogram.1252

And then, if a parallelogram is like this, then it is a rectangle.1256

In the same way, I can maybe use, let's say, dogs.1260

Now, if I say "animals," saying "animal" is like saying "quadrilateral"; it is very general; it is just very broad.1266

And then, a type of animal would be, let's say, "dogs."1280

And then, a more specific type would be, let's say, a Chihuahua.1288

This is the same type of concept.1303

Now, using that, let's look at these problems again: if a quadrilateral is a rectangle, then it is a parallelogram.1305

So, if an animal is a Chihuahua, then that Chihuahua is a dog.1317

Is that true? Is that always true, sometimes true, or never true?1323

This would be always; if it is a Chihuahua, then it is always a dog; Chihuahuas are always dogs.1326

So then, this is "always."1333

If a quadrilateral is a parallelogram, then it is a rectangle; now, are parallelograms always rectangles?1339

No, sometimes they are just parallelograms; in the same way, if the animal was a dog, then it is a Chihuahua.1350

Well, dogs can be something else; there are different types of dogs.1362

There is the golden retriever; there is the Maltese; there are all of these different dogs.1366

The Chihuahua is just one type; so just because it is a dog doesn't mean that it is always going to be a Chihuahua.1372

But it can be; so then, this one would be "sometimes."1377

If a parallelogram has a right angle, then it is a rectangle.1385

If a parallelogram has a right angle, then automatically, a right angle is a property of a rectangle.1392

Now, even though rectangles say that there are four right angles, just the fact that there is one in a parallelogram--1398

that means that all four have to be right angles, because from this, we know that if there is just one,1405

well, opposite angles are congruent; that means that this one has to be one, too.1418

Well, aren't consecutive angles supplementary?--so, if this is a right angle, if this is 90 degrees,1422

180 - 90 is 90, so this has to be 90; and then, this is also the same as that, so...1428

If there is one right angle in a parallelogram, then it is a rectangle--then all four would be right angles.1436

Now, if they said, "If a quadrilateral has a right angle, then it is a rectangle," that is not going to be "always."1444

That is actually going to be "sometimes."1452

A quadrilateral with one right angle is very possible; that means that we can have something like that, where this is the only right angle.1454

So, this is a quadrilateral, and this is a parallelogram.1467

A parallelogram with a right angle would make it a rectangle.1471

A quadrilateral with a right angle is not always going to be a rectangle.1477

Then, this would be "always," because they said "parallelogram."1483

The last one: If opposite angles of a quadrilateral are congruent, then it is a rectangle.1491

Now, notice how they say the word "quadrilateral"; they don't say "parallelogram."1505

Can you draw a counter-example? Remember: this is a counter-example; it is an example showing the opposite.1513

If opposite angles of a quadrilateral are congruent, can you draw a quadrilateral with opposite angles being congruent, but it is not a rectangle?1523

Yes, I can; how about that? Opposite angles are congruent, and that is a quadrilateral.1534

So, opposite angles of a quadrilateral are congruent, but it isn't a rectangle.1546

It could be, because I could also draw something like this, where opposite angles are congruent, which would just be like this.1553

Or if you want, just show that they are congruent and opposite.1566

Then, either way, these both are true for #4.1571

This would be "sometimes," meaning that opposite angles of a quadrilateral can be congruent, but it doesn't always have to be a rectangle.1577

Sometimes, it could be, in this case; and it is not in this case; so it is sometimes.1597

And the last one, the fourth example: We are going to determine if ABCD is a rectangle, given all of their vertices.1606

Again, we want to find their slope, because we know that rectangles have perpendicular sides.1616

We know that perpendicular lines have negative reciprocals of each other--their slopes are negative reciprocals of each other.1626

So, we are going to find the slopes of each of the lines, and then see if they are negative reciprocals.1636

Just to draw a little rectangle: this is A, B, C, and D.1645

You know what sides have to be perpendicular with what other sides.1651

The slope of, let's say first, AB: now, if you want to do this problem, you have to know the formula for slope.1655

So, slope is the difference of the y's, y₂ - y₁, over the difference of the x's.1666

Here we are going to use this point and this point.1678

The difference of their y's would be 6 - 5, over -2 - 3, which is going to be 1 over...this is -5.1680

Then, I am going to find the slope of BC: 0 - 6, over 2 - -2.1696

This is going to be -6 over...this is 4; and then, that is going to be -3/2.1713

Now, AB has a slope of -1/5, and BC has a slope of -3/2.1727

So, they are not perpendicular, because their slopes are very different; they are not negative inverses, or reciprocals, of each other.1740

Therefore, this is not a rectangle; so then, this is "no--not a rectangle."1750

That is it for this lesson; we are going to cover other types of parallelograms.1766

We are going to go over the square and the rhombus.1773

And then, after that, we are going to go over the trapezoid and kites.1780

So, we are going to go over different types of parallelograms next.1784

Thank you for watching Educator.com.1786