Geometry Squares and Rhombi

Welcome back to Educator.com.0000

In the next lesson, we are going to continue on with parallelograms.0002

And we are going to go over, more specifically, squares and rhombi.0005

First, the rhombus: now, rhombus and rhombi are actually the same thing.0011

Rhombus is singular, so when you have only one, then it is a rhombus; when it is plural--you have more than one--then it is actually called rhombi.0018

So then, if you hear "rhombus" or "rhombi," then you are talking about the same thing.0028

Now, what is a rhombus? A rhombus is a quadrilateral with four congruent sides.0034

Here is an example of a rhombus: now, more specifically, a rhombus is a special type of parallelogram.0043

You can say that a rhombus is a parallelogram with four congruent sides.0050

This right here, this property, is very specific to the rhombus.0055

Now, to continue our flowchart, if we have a quadrilateral, quadrilateral goes down to parallelogram;0063

and then, parallelogram...we went over the rectangle; that was a special type of parallelogram;0079

and then now, we are going to go over the rhombus: it is another special type of parallelogram.0089

Now, because the rhombus is a special type of parallelogram, all of the properties of a parallelogram now apply to the rhombus,0100

just like when we went over rectangles: since a rectangle is a special type of parallelogram,0110

all of the properties of parallelograms apply to the rectangle.0115

The same happens with a rhombus; all of the properties apply to the rhombus, as well.0118

We went over one property that is very specific to the rhombus; it is all of the properties of the parallelogram, plus four congruent sides.0127

Now, there are also a couple of properties that have to do with the diagonals of a rhombus.0136

The first one: the diagonals of a rhombus are perpendicular.0142

Now, the property on diagonals for a parallelogram is that they just bisect each other.0147

They bisect each other; so we know that diagonals bisect each other--that is the very general property of the parallelogram.0153

Then, for the rectangle, it became that they bisect each other, and they are congruent.0173

And then now, the diagonals of a rhombus bisect each other and are perpendicular.0181

Let's say I have a rhombus that...here are my diagonals; we know that diagonals bisect each other.0193

That means that this diagonal and this part are congruent.0205

And then, the other two halves are congruent to each other.0209

And then, this one, the one that is a little more specific to the rhombus, is that they are also perpendicular.0213

So, if it is a rhombus, then the diagonals are perpendicular.0223

And this is the theorem that goes with that; so if the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.0229

This property is very specific to the rhombus, meaning that this property only applies to the rhombus--0241

so much so that you can actually use it to prove that it is a parallelogram.0248

So, if you can prove that the diagonals are perpendicular, then you can prove that that parallelogram is a rhombus.0253

Any time you see anything that is perpendicular--diagonals being perpendicular--you know automatically that that is a rhombus--0263

of course, as long as it is a parallelogram (it is a parallelogram with diagonals perpendicular--then, a rhombus).0269

The next property that has to do with the diagonals of a rhombus is that they bisect opposite angles.0277

There is my rhombus; there are my diagonals; we know that it is perpendicular; we know that the sides are congruent.0296

But then, this one now says that the diagonals bisect opposite angles.0308

So then, it is bisecting those angles, meaning that this angle is now cut into half.0316

And then, these angles are bisected, and these angles are bisected, so that, when they bisect a pair of opposite angles,0325

the opposite angles are also congruent, because we know that opposite angles are congruent,0336

because that is a property of a parallelogram: Opposite angles are congruent.0340

This whole angle is congruent to this whole angle; so this is cut in half, and each of these halves are also congruent to each other.0345

Again, there are two more properties of the rhombus, in addition to all of the properties of a parallelogram.0355

#1: Diagonals are perpendicular; #2: Diagonals bisect opposite angles.0362

Using those properties, let's talk about finding the missing value.0379

Again, here is our rhombus; and you know that angle 1 is congruent to angle 2,0387

because this diagonal bisects the angles, because it is a rhombus.0403

That means that I can just make the measure of angle 1 equal to the measure of angle 2: 2x + 6 = 3x - 19.0410

I am going to solve for x by subtracting the 2x over to the other side; I am going to get 6 = x - 19.0425

I am going to add 19 to the other side, and I get 25 = x; so there is my x.0433

Now, if I need to find the measure of angle 1, I am just going to plug it in; so the measure of angle 1 equals 2(25) + 6.0442

So, the measure of angle 1 is...this is 50, plus 6 is 56.0456

Now, you know that the measure of angle 1 is 56; the measure of angle 2 is also going to be 56.0462

But just to check your answer, you can find the measure of angle 2: 3 times...x is 25...minus 19.0469

3 times 25 is 75, minus 19...that is 56.0482

So then, we know that our answer is correct.0492

Now, next is the square: squares are actually very, very special.0498

How are they special? If you look at the definition, a quadrilateral with four right angles and four congruent sides,0505

we already heard these properties--we have already talked about these properties before.0516

If we break it down, four right angles--what has that property of four right angles?0524

We know that that belongs to the rectangle; this is the rectangle's property, four right angles.0534

And then, a quadrilateral with four congruent sides--that one belongs to something else, too, and that is the rhombus.0544

A rhombus is a quadrilateral with four congruent sides.0555

So, that means that a square is made up of the rectangle and the rhombus, which is why a square is a special type of both.0559

It has the properties of both the rectangle and the rhombus.0570

Continuing with our little flowchart: a quadrilateral goes down to a parallelogram, and then a parallelogram...0575

we went over two types, the rectangle and...we just went over...the rhombus;0595

and we know that a square is a special type of rhombus and rectangle; and that is the square.0606

Remember that a rectangle has all the properties of a parallelogram, plus its own.0622

And then, a rhombus has all of the properties of a parallelogram, plus its own properties.0633

So then, the square has all of the properties of a rectangle (since it is a type of rectangle);0642

it has all of the properties of a rhombus, which means that it also has all of the properties of a parallelogram.0650

So, a square is made up of all of these above; a square is just a special type of everything.0656

Now, here is that chart again; but let's actually go over each of the properties.0667

We know that a parallelogram is two types, rectangle and rhombus; and then, a square is a type of rectangle and rhombus.0674

But what about their properties?--let's go over their properties again.0683

A parallelogram, we know, has...the definition of a parallelogram says...two pairs (let me write that out) of opposite sides parallel.0686

And this is more of the definition of a parallelogram: two pairs of opposite sides parallel.0711

The properties: two pairs of opposite sides are congruent; two pairs of opposite angles are congruent;0717

and the diagonals bisect each other; and then, one more--consecutive angles are supplementary.0738

That is everything that has to do with a parallelogram.0766

Now, the rectangle: we know that a rectangle has...0772

Instead of listing all of these out, because we know that rectangles have all of the properties of a parallelogram,0793

instead of writing all of these out, we will just say "all properties of parallelogram."0798

That means that it includes all of this.0805

We know that it has four right angles, and then, what about their diagonals? Diagonals are congruent.0809

Those are the properties of a rectangle.0831

For the rhombus, again, it has all properties of a parallelogram, and then, four congruent sides;0835

and then, their diagonals are perpendicular, and the diagonals bisect the angles.0862

Those are the properties of a rhombus; now, what about a square?0882

I know that a square has all of the properties of a parallelogram, so I am going to write that out: "all of parallelogram;0888

all of the properties of a rectangle"; and then, "all properties of the rhombus."0899

A square doesn't really have anything that is specific to its own; it just takes on all of the properties of everything else; that is a square.0916

That is why it is a little bit special; it is just a mixture of everything.0925

That is a summary chart; now, above this, we know that it is a quadrilateral, just a four-sided figure;0932

and then, it is more specific to the parallelogram, and so on down.0938

Let's now go into our examples: for our first one, we have this table, and all of the properties of diagonals.0946

So, you are going to write "yes" or "no" in each of the boxes, depending on if the diagonal property applies to that type of quadrilateral.0956

We have our parallelogram, rectangle, rhombus, and square.0971

The first property: The diagonals bisect each other.0975

Now, try to remember what property that is for--"the diagonals bisect each other."0982

That is a property of a parallelogram; so this one would be "yes."0989

Now, if that one is "yes," that means that this one applies to all of the other ones, because rectangles have all of the properties0998

of a parallelogram; so does a rhombus, and so does a square; this would be "yes," "yes," and "yes."1006

Now, the next one: Diagonals are congruent.1018

Well, I know that not all parallelograms have congruent diagonals; so then, this would be "no."1023

Or you can just leave it blank, just so you can see which ones are actually "yes"; it is just easier to see.1031

Let's look at this: what about rectangles--do rectangles have congruent diagonals?--and this one is "yes."1040

Does the rhombus have congruent diagonals?--this one is "no"; it does not.1050

The diagonals of a rhombus are perpendicular to each other, and they bisect the angles, but they are not congruent to each other.1057

That is not a property of a rhombus.1067

Now, if you ever get confused, just draw it out; and you know that, if you had to walk from here to here,1069

that just seems a lot further than walking from here to here; the distance looks a lot longer this way than it does this way.1080

So, they don't seem congruent, and the same thing with a parallelogram; so then, this would be "no."1086

But what about a square? Squares have all of the properties of the rectangle.1093

So, whatever applies to the rectangle also applies to the square, so this one is "yes."1099

Then, the next one: Each diagonal bisects a pair of opposite angles, meaning it does this.1106

Now, we have only gone over this one time, because it is very specific to one thing, and that is the rhombus; it is a property of the rhombus.1123

Now, again, squares have all of the properties of the rhombus; so then, this will also be "yes."1135

And then, the last one: The diagonals are perpendicular--which one is this for?1144

This one is also for the rhombus; the rhombus had both of these two diagonal properties, so this one is "yes";1151

and again, squares have all of the properties of the rhombus, so this one is also "yes."1158

And then, all of the blanks will just be "no"; so you can just fill it in with "no" if you have it written down.1164

Otherwise, this way, you can just see what is "yes" and what is "no."1170

OK, the next example: we are going to use the rhombus ABCD to find the missing value.1177

We have two problems; this is a problem, and this is another problem.1186

The first one: The measure of angle ABC is 120; find the measure of angle ACB.1193

We are given that this one is 120, and we don't know this one right here; this is what we are looking for, right here--this little part, angle ACB.1205

Let's see: we know that this is 120; now, we can do this a couple of ways.1219

We can say that, since this is 120, we can use a property of the parallelogram, because a rhombus has all of the properties of a parallelogram.1229

We can use the property of a parallelogram that says that consecutive angles are supplementary.1241

So then, if angle ABC is 120, then that would mean that angle BCD is the supplement to that; so that would be 60.1250

This whole thing is 60; and then, since it is a rhombus, we know that the diagonals are perpendicular, and that it bisects the angle.1262

These are cut into two equal parts; so if this whole thing is 60, then this has to be 30--it has to be half.1274

The measure of angle ACB is 30 degrees.1283

Now, another way to find that is to say that, if this whole thing is 120, then this would be 60;1288

this is a right angle; then, this is a triangle, so this would be a 30-60-90 degree triangle,1298

because the three angles of a triangle add up to 180.1308

A 30-60-90 triangle is a special right triangle; you just know that this is 90, so that means that these two angles have to be 90.1313

That is another way to find this angle measure.1323

Now, the next one: The measure of angle AED, this one right here, is 4x + 10; find x.1327

That is all that they are going to give you--that it is 4x + 10.1345

But since we know that this is perpendicular (remember: the diagonals are perpendicular), this angle measure is 90.1350

So, make 4x + 10 equal to 90, since that is the angle measure here; and then, we know that it is equal to 90, so 4x = 80; x = 20.1357

The next example: Determine if each statement is always, sometimes, or never true.1387

If all sides of a quadrilateral are congruent, then it is a square.1394

Well, all of the sides being congruent--that is a property of a rhombus.1409

And a square is a special type of rhombus, but not all rhombi are squares.1421

Sometimes a rhombus is just a rhombus; it doesn't have to always be a square, so this one would be "sometimes."1427

If all of the sides of a quadrilateral are congruent, then it is always a rhombus; but a rhombus is sometimes a square.1439

If a quadrilateral is a parallelogram, then it is a rhombus.1450

Well, a parallelogram is not always a rhombus; it is sometimes a rhombus, because a parallelogram could also be a rectangle.1457

So, a parallelogram is only sometimes a rhombus.1470

The next one: If a quadrilateral is a rhombus, then it is a parallelogram.1480

Well, yes, because a rhombus is always a parallelogram--it is a special type of parallelogram.1485

Just like we said, if we have a dog, a type of dog is, let's say, Chihuahua.1489

Well, a Chihuahua is always a dog; since a Chihuahua is a type of dog, a Chihuahua is always going to be a dog.1495

Just like that, a rhombus is a special type of parallelogram; so a rhombus is always a parallelogram; so this is going to be "always."1506

The last one: If a quadrilateral is a square, then it is a rectangle.1517

Is a square always a rectangle? It is almost the same as #3: a rhombus is a type of parallelogram; therefore, it is always a parallelogram;1523

a square is a special type of rectangle; therefore, a square is always a rectangle.1532

Now, if this is still confusing, we can use that flowchart that we did to help us with this.1540

If I say, let's say, "parallelogram," instead of writing it all out--we know that that is a parallelogram--1549

a parallelogram became a rectangle, and it became a rhombus; then this and this became a square; on top of that is a quadrilateral.1559

Now, if it is going from "if" to "then" downwards, then the answer is going to be "sometimes."1574

If it is going from "if" to "then" upwards, then it is "always"; and if it is going side-to-side, then it is "never."1588

Let's look at these again: #1: If all sides of a quadrilateral are congruent, then it is a square.1598

We know that they are talking about the rhombus, basically saying that, if it is a rhombus, then it is a square.1605

Well, isn't this sometimes? A rhombus is only sometimes a square; a rhombus can just stay a rhombus, so that is "sometimes,"1613

because it is going downwards, from a rhombus to a square.1619

The next one: If a quadrilateral is a parallelogram, then it is a rhombus.1624

See how it is going downwards from a parallelogram to a rhombus; so if it is going down, it is "sometimes."1628

Down is sometimes; up is always; and side-to-side means never.1637

If a quadrilateral is a rhombus, then it is a parallelogram.1655

It is going upwards; then it is always, because this is a special type of that.1660

And then: If a quadrilateral is a square, then it is a rectangle.1666

See how it is going upwards? That is always; so you can use this to help you with this.1671

The next example: Determine whether quadrilateral ABCD is a parallelogram, a rectangle, a rhombus, or a square with the given vertices.1684

With these vertices, we are going to have to do a few things to see what the most specific type of quadrilateral is.1696

We know that, first of all, to figure out if it is a parallelogram, we have to find the slope.1709

Slope will help us with anything that has to do with being parallel or perpendicular, which has to do with both of these, this one and this one.1718

And then, for the rhombus, as long as we can show that it is a rectangle and a rhombus,1732

then we can just automatically say that it is a square, too, if it is both a rectangle and a rhombus.1742

Now, how do you show if it is a rhombus?1749

Well, you know that a rhombus has four congruent sides, so we would just have to use the distance formula to show the length of each side.1752

If parallelogram works, then we have to continue with rectangle; if rectangle works, we are going to continue with rhombus.1760

If this doesn't work, then it would be a rectangle; but let's say rectangle and rhombus worked--then it would be a square,1768

because "square" means that it is both rectangle and rhombus.1777

Oh, and then, also, for a parallelogram, if rectangle doesn't work after parallelogram, then we would move on to rhombus.1783

We would have to actually show all three of these, but we don't have to show the square, because the square is just all of the above.1791

The slope of AB: again, it is the difference of the y's (so I am going to take the first y, which is -6,1802

and subtract the second y), and then take the first x, and subtract the second x.1812

A minus negative: this becomes a plus; this is -3 over 4.1827

Then, if you need help to determine which sides have to be parallel to what, and what sides have to be perpendicular to what,1837

just draw any quadrilateral, ABCD; you know it has to be in this order.1844

It doesn't matter if you do ABCD or if you do ABCD...it doesn't matter, as long as you know that it is in the order.1849

Then, the slope of BC: -3 minus -6, over 5 minus 1: this, again, becomes positive; this is 3/4.1860

Well, that is strange; let's see--let's double-check this.1888

For AB, -6 - -3 became -3; 1 - -3 became 4; then for this one, y₂ is -3, minus y₁, which is -6;1897

and then, x₂, which is 5, minus x₁, which is 4...this is 3/4; so it seems like it is correct.1923

Now, here, this slope is -3/4, and this slope is positive 3/4.1938

That means that they are not perpendicular to each other, because, in order for this side and this side to be perpendicular,1948

their slopes have to be the negative reciprocal of each other.1957

Then, if this is going to be -3/4, then this has to be positive 4/3, but it is not.1960

So, automatically, I can conclude that this AB and this are not perpendicular.1969

If it is not perpendicular, then I can cross out some of these: I can cross out the rectangle, and I can cross out the square,1978

because in order for it to be a square, this has to be true; rectangle has to be true, and rhombus has to be true.1986

Now, let's continue, because we still have two other options.1993

I am going to go for my next one, the slope of CD: 0 - -3 over 1 - 5.2000

So again, this becomes 3/-4; so that means that AB is parallel to CD.2013

Does that make that a parallelogram?--no, not yet: it has to be two pairs of opposite sides being parallel.2027

Or we can say that AB is congruent to DC, if you remember from the section on the parallelogram--2033

on the properties, or the theorems to prove a parallelogram.2040

There is one theorem; it is not a property of a parallelogram--it is a theorem that says that,2045

if you can prove that one pair is both parallel and congruent, then it is a parallelogram; that is another way you can do it.2051

Now, the last one, CB and then AD: 0 - -3, over 1 - -3, becomes 3/4.2065

So then, BC and AD are parallel, because they have the same slope.2084

So automatically, you know that that is a parallelogram; but then, a rhombus is also considered a parallelogram.2091

So then, we have to find the distance of each one, the distance of AB (let me do a lowercase d)...2096

We know that the distance formula is (x₂ - x₁), the difference of the x's, squared,2113

with the difference of the y's, squared, all under a square root.2125

For AB: 1 - -3, squared, plus -6 - -3, squared; this is going to be the square root of...2133

here is 4 squared; that is 16, plus...this is -3, squared, which is 9; that is going to equal √25, which is 5.2156

And then, let's find BC: the distance of BC: this is 5 + 6, squared, plus -3 + 6;2173

and I am just making them automatically plus for minus a negative.2200

Oh, wait: 5 - 1...I did the y instead of the x...the 5 minus the 1 is going to be 4 squared, which is 16,2207

plus 3 squared, which is 9, which is equal to √25, and that is 5.2226

And then, we know that this AB is going to be congruent to BC.2233

Let's try the other ones: the distance of...what is next?...CD: the square root of 5 - 1, squared...2245

Now, notice how I did this one minus this one before I did this one minus this one--it is the same thing,2263

as long as, for the next part, for the y's, you do the same order.2269

If you are going to do 5 - 1, then for the y's, you have to do -3 - 0, squared.2276

That is going to be right here...4² is 16, plus...(-3)² is 9.2285

So again, that is the square root of 25, which is 5.2293

And then, the distance of the last one, AD: -3 - 1, squared, plus -3 - 0, squared: here we have -4 squared, which is 16,2298

plus -3 squared, which is 9; so then, this is the square root of 25, which is 5.2321

So, we know that all of the sides are congruent; that means that we have a rhombus, so this is a rhombus.2329

That is it for this lesson; thank you for watching Educator.com.2350