Geometry Proving Parallelograms

Welcome back to Educator.com.0000

For this lesson, we are going to use the theorems and the properties you learned in the previous lesson to prove parallelograms.0002

Turning the properties that we learned into actual theorems, if/then statements:0012

the first one: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.0020

Now, these theorems have no name; we have no name for the actual theorem, so we actually have to write it all out.0030

If I say, "If opposite sides are congruent, then it is a parallelogram," you can shorten it in that way.0038

So, if you ever have to use this theorem on a proof, then you can just shorten this as your reason,0060

instead of having to write this whole thing out; "if opposite sides are congruent, then it is a parallelogram."0066

Do something like that; you can just shorten words and phrases.0071

Then, our conditional statement: as long as we have opposite sides being congruent...if this, then parallelogram.0076

And this just means "parallelogram"; or actually, I can write it all out; maybe that will not be as confusing: "then parallelogram."0097

The second one: "If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram."0113

As long as we have (just like the property we learned in the previous lesson) a parallelogram, then we know that both pairs of opposite angles are congruent.0122

In the same way, the converse would be, "If both pairs of opposite angles are congruent, then it is a parallelogram."0134

It is just basically saying that if the opposite angles are congruent, then it is a parallelogram.0162

So, as long as we can prove this or this, then we can prove that it is a parallelogram.0180

Now, we have other options, too; there are actually more theorems.0186

The third theorem that we can use to prove quadrilaterals parallelograms is on their diagonals.0191

If we can prove that the diagonals (you can just say "if diagonals") bisect each other, then it is a parallelogram.0202

You can shorten it in that way; if you can just prove that the diagonals bisect each other,0223

in that way, then you have proven that the quadrilateral is a parallelogram.0235

Oh, I had it right...parallelogram.0250

And the fourth one, the last one: "If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram."0255

This is one that wasn't on the previous lesson; this is actually not a property of a parallelogram.0267

This is just an extra theorem that says that if you can prove that only one pair of opposite sides is both,0279

parallel and congruent, then you can prove that it is a parallelogram.0295

Now, again, this is not a property of a parallelogram; it is just that you have to prove that one pair of opposite sides is both parallel and congruent.0310

That is one way that you can prove that it is a parallelogram.0320

With other theorems, you have to prove two pairs: the first one was two pairs of opposite sides being congruent;0325

the second one was two pairs of opposite angles being congruent; for this one, you have to prove that both diagonals bisect each other.0332

But for this one, this is the only theorem where it has one pair, but it just has to be two things about that one pair of sides.0341

So then, you can just shorten it by saying, "If one pair of opposite sides is parallel and congruent, then it is a parallelogram."0353

Maybe you can say something like that--just shorten it like that, in that way.0375

This right here--we are just determining if this quadrilateral is a parallelogram.0383

In the previous lesson, we did a couple of these; in that case, the problems before in the last lesson,0390

you knew that it was a parallelogram, but then you just had to show that the slopes are the same, show that the sides were congruent...0399

For this problem, we have to determine if it is a parallelogram.0409

We don't know that it is a parallelogram; so then, using the same methods, using the distance formula,0416

we have to see if it is going to come out to be the same.0421

If these two are the same, and these two are the same, then we have to say that it is a parallelogram.0424

So, it is the same thing; you are using the same methods.0434

Before, all you were doing was just showing the numbers of the parallelogram, showing that this is 5, and this is 5, too, and so on.0437

And that is it--just verifying; you were just giving the measurements of them.0448

But for this, we are actually proving that it is a parallelogram by finding distance or finding slope and seeing whether or not they are the same.0452

Again, you can use the distance formula, or you can use slope.0464

If you are going to use the distance formula to show that these opposite sides are congruent,0469

and that these opposite sides are congruent, then you are going to be using the first theorem we went over,0474

saying that if two pairs of opposite sides are congruent, then it is a parallelogram.0479

If I use slope and find the slope of AB, find the slope of CD, and they are the same, that is showing that they are parallel.0485

And then, I find the slope of AD and the slope of BC, and say that they are the same--they have the same slope, which means that they are parallel.0495

I am not using one of the theorems, because remember: we said that if you state that two pairs of opposite sides are parallel,0504

that is just the definition of a parallelogram; so by definition, we can say that it is a parallelogram, if we use slope,0516

because then we are showing that opposite sides are parallel.0523

We are not using one of the theorems; we are actually just using the definition of a parallelogram.0526

It doesn't matter which one you use; you can just use one of the theorems, or you can use the definition of parallelogram to show that they are parallel--whichever.0531

And then, the distance formula, if you wanted to use that, is the square root of0541

the first x minus the second x, squared, plus the first y minus the second y, squared.0548

Slope is y₂ - y₁, over x₂ - x₁, or rise over run.0558

Rise measures up/down; run measures left/right.0573

In this case, slope will probably be a little bit easier, because for slope, all you have to do is count.0580

You can just count how many units you are going up, down, left, and right, whereas with distance, you have to calculate each thing out.0587

This also: if you have the points written out for you, then this can be pretty easy.0597

But we are just going to use the rise and run to find the slope by counting.0606

When you move up, that is a positive number, and that is going to go on the top, in the numerator.0614

When you go to the right, it is a positive; when you go down, it is a negative; and when you go to the left, it is a negative.0622

So then, that is because when you go up, you are going towards the positive y-axis.0628

If you to the right, you are going towards the positive x-axis.0634

If you go down, then you are going towards the negative y-axis; you are going towards the negative numbers, so if you go down, it is a negative number.0637

If you move left, you are going towards the negative x numbers, so that is also a negative number.0644

From A to B: now, it doesn't matter if you travel from A to B, or if you go from B to A--it does not matter.0652

So, if we go from A to B, we are going to count up 3; remember: going up is positive, so that is positive 3, over...0659

we go to the right 1, so the slope is 3/1, or just 3. The slope of AB is 3.0670

For BC, I am going to count from B to C; so I am going to count up/down first, the rise; do that one first.0683

From B to C, I have to go down; I am going to go 1, 2, 3, 4; I have to go down 4; so the slope of BC is -40692

(because going down is negative)...then from here, I am going to go 1, 2, 3, 4.0704

So, I went to the right 4, and that is a positive, because I went to the right, which makes this slope -1.0710

From C to D (it doesn't matter if you go from D to C or C to D), if I want to go from C to D,0720

then I am going to count 1, 2, 3, down 3; so the slope of CD is down 3, which is -3, over...0726

from here, I am going to go left 1; left 1 is -1; so then, -3/-1 is 3.0737

And then, from D to A, I can go...the slope of AD is 1, 2, 3, 4; that is a positive 4, because I am going up 4;0749

then 1, 2, 3, 4...that is a negative 4; I am going to the left 4.0764

And that makes this a negative 1; so since AB and CD have the same slope, I know that AB is parallel to CD.0770

And BC and AD have the same slope; that means that they are also parallel.0794

So, BC is also parallel to AD; I have two pairs of opposite sides parallel.0802

So, by the definition of parallelogram, this is a parallelogram, so yes, quadrilateral ABCD is a parallelogram.0813

OK, let's just summarize over the different theorems that we can use to prove parallelograms, before we actually start our examples.0843

A quadrilateral is a parallelogram if any one of these is true.0856

You don't have to prove all of these; just prove one of them.0863

If you prove one of these, then you can prove that the quadrilateral is a parallelogram.0867

The first one: a quadrilateral is a parallelogram if both pairs of opposite sides are parallel.0873

That is the definition of parallelogram; so as long as you can prove (this is the definition of parallelogram)--0882

as long as you can show--that this side is parallel to this side, and this side is parallel to this side,0892

then by the definition of parallelogram, the quadrilateral is a parallelogram.0902

The second one: If both pairs of opposite sides are congruent...as long as you show0909

that this side is congruent to that side and this side is congruent to that side, then you can state that this is a parallelogram.0916

Both pairs of opposite angles are congruent: that means that this angle is congruent to this angle, and this angle is congruent to this angle.0925

And remember: it has to be two pairs of opposite angles being congruent.0937

Then, that is a parallelogram.0940

Diagonals bisect each other--not "diagonals are congruent," but "they bisect each other."0945

That means that this diagonal is cut in half, and this diagonal is cut in half.0953

Those two halves are congruent; then this is a parallelogram.0958

And then, this is the one that is a little bit different; we have seen these as properties, but the last one is a special kind of theorem0966

that says, "Well, if you can prove that one pair of opposite sides (it doesn't matter if it is this pair or this pair,0980

as long as you can prove that that one pair of opposite sides) is both parallel and congruent, then this will be a parallelogram."0986

So, if you have to prove parallelograms, you can just use any one of these five--whichever one you can use, depending on what you are given.0997

Then, you can do that to prove parallelograms.1006

Let's actually go through some examples now: the first one: Let's determine if each quadrilateral is a parallelogram.1012

In this case, the first one, I have one pair of opposite sides being parallel, and I have the other pair of sides being congruent.1022

Now, if you remember, from the theorems and the definition of parallelogram that we went over, none of them say that this is a parallelogram.1034

So, if I see that one pair of opposite sides is parallel, and the other side is congruent, that is not a parallelogram.1045

This could be a parallelogram, but there is no theorem, and there is no definition, that says this.1056

The closest one...well, there are a few; one of them says that it has to be both pairs of opposite sides being parallel.1063

We have one pair being parallel; if these two sides were parallel, then we could use the definition of parallelogram.1073

If both pairs of opposite sides are congruent...we have one pair that is congruent; this pair is not congruent, so then we can't use that.1079

And then, the last one, the special one that we went over--that has to be the same pair.1089

So, one pair, the same pair of opposite sides, being both parallel and congruent--then it is a parallelogram.1095

So, if these sides are both parallel and congruent, then we have a parallelogram.1104

Or these sides--if they were both parallel and congruent, then we can use that one; but it is none of those.1111

So, this one is "no"; we cannot determine it.1121

It could be a parallelogram, but we can't prove it, because there is no theorem--nothing to use to state as a reason, so this is a "no."1125

The next one: all I have are four right angles--nothing else; just four right angles.1135

Now, for this one, the theorem that has to do with angles is "opposite angles are congruent."1143

Now, this angle and this angle--are they congruent? Yes, they are.1151

This angle and this angle--are they congruent? Yes, they are.1156

So, this, therefore, is a parallelogram, so this one is "yes."1161

We can use that one theorem that says that two pairs of opposite angles are congruent.1166

Now, some of you are probably looking at this and thinking, "But that is a rectangle!"1173

Yes, it is a rectangle; we are actually going to go over rectangles next lesson.1177

But a rectangle is a special type of parallelogram, so rectangles are parallelograms.1183

So, is this a parallelogram? Yes, it is.1193

So, if we have a rectangle, then we have a parallelogram.1198

But then, without even thinking of rectangles, with this alone, just looking at the angles, opposite angles are congruent;1204

so we have two pairs of opposite angles being congruent.1212

By that theorem, we have a parallelogram; so this is "yes."1218

All right, the next one: Find the value of x and y to ensure that each is a parallelogram.1225

ABCD: now, we have to be able to find the value of x and y so that these two sides will be congruent, and then these two sides will be congruent.1233

If I want these two sides to be congruent, and find a number for y that will make these congruent, then I have to solve them being congruent.1246

5y is equal to y + 24; here, I am going to subtract the y; so that way, this will be 4y is equal to 24.1257

Then, I divide the 4 from each side, and y is equal to 6.1273

If y is 6, then this will be 30; if y is 6, then this will be 30; so then, that is the value for y.1280

And then, for x, again, I have to do the same thing: so, 2x + 3 is equal to 3x - 4.1289

I am going to subtract the 2x here; you can add the 4 to this, so 7 is equal to x.1299

If x is 7, then this will be 14; this will be 17; here, this is 21 - 4, is 17.1317

The next one: now, this looks like it would be a square or rectangle, but you can't assume that.1329

I don't have anything that tells me that these are right angles; I don't have anything that tells me anything, really.1338

I have to find x and y so that these diagonals will be bisected, because that is what I am working with.1346

Then, this and this have to be congruent; this and this have to be congruent.1353

I am going to make x + 1 equal to 2x - 3, and then subtract the x here; 1 = x - 3; add the 3; then, this is 4 = x.1361

Let's see, the y's: this y and then this...this one is y + 4 = 20 - 3y.1388

So, if I add 3y, this is 4y + 4 = 20; subtract the 4; 4y = 16; divide the 4, and y = 4.1405

So then, now, if x is 4, and y is 4, then these parts of the diagonals will be congruent.1426

Therefore, the diagonals bisect each other, and then as a result, this is a parallelogram.1437

The next example: Determine if the quadrilateral ABCD is a parallelogram.1446

We are given the coordinates of all of the vertices of the quadrilateral.1453

And then, we have to determine if it is a parallelogram.1460

I can just draw it out here; it doesn't matter how you draw it, as long as, remember, when we label this out,1467

it has to be ABCD, or vertices have to be next to each other; it can't be jumping over, so it can't be ACDB--none of that.1475

It has to be in the order, consecutive.1488

And I am drawing this just to show which coordinates are next to each other, which ones are consecutive.1493

Again, you can use slope, or you can use the distance formula.1502

Since we used slope last time, let's use the distance formula this time.1506

I am going to find the distance of AB, compare that to the distance of CD, and see if they are congruent.1510

And then, before you move on, why don't you just try those two and see if they are congruent,1518

because if they are not, then you don't have to do any more work; you can just automatically say, "No, it is not a parallelogram."1523

So, just do one pair of sides first; and then, if they are congruent, then move on to the next pair, and then see if they are congruent.1531

The distance formula is (x₁ - x₂)² + (y₁ - y₂)².1544

The distance of AB is the square root of 5 - 9, squared, plus 6 - 0, squared.1561

5 - 9 is -4, squared; plus 6 squared...this is 16 + 36, which is 52.1576

Now, you can go ahead and simplify it if you want.1596

Your teacher might want you to simplify it.1599

But since all we are doing is just comparing to see if AB and CD are going to be the same,1602

I can just leave it like that, and then see if CD is going to come out to be the same thing.1608

If your teacher wants you to actually find the distance of each side and show the distance,1614

and make it simplified or round it to the nearest decimal, then you have to simplify that.1620

Or else, if it is just to determine if it is a parallelogram, then you can just leave it.1629

A way to simplify that, though, just to show you: we know that 52 is not a perfect square.1634

So, what you can do is a factor tree: 52...2 is a prime number, and 26; 2--circle it--and 13.1642

So, this is the same thing as the square root of 2 times 2 times 13; and then, we know that this can come out as a 2.1658

So then, this is 2√13.1668

CD next: CD is, using these two, 8 - 3 squared, plus -5 - 2, squared.1678

Oh, -5 minus a -2...that is a plus.1698

And then, the square root of...this is 5 squared, plus -3 squared; 25 + 9...this is 34.1702

We found AB and CD, and they are not the same; let me just double-check.1722

Let's double-check our work; this is 5 - 9, squared; 6 - 0, squared.1730

And then, for CD, it is 8 - 3, squared, and -5 - -2, squared.1740

We have 16 + 36, which is 52, so √52; the square root of 5 squared plus -3 squared is 25 + 9, which is √34.1752

So, I know that, since these are not congruent (this is √52, and this is √34), they are different.1767

I can stop here; I don't have to continue and show my other two sides (again, unless your teacher wants you to).1782

If all I have to determine is if this is a parallelogram or not, then I can just stop here and say, "No, it is not a parallelogram."1790

No, quadrilateral ABCD is not a parallelogram, because opposite sides are not congruent.1801

If it was congruent, if they were the same, then you would have to go ahead and find the distance of BC,1818

find the distance of AD, and then compare those two.1823

For the last example, we are going to complete a proof of showing that it is a parallelogram.1829

Always look at your given; using your given, you are going to go from point A1840

(this is your point A; this is your starting point, and then this is your ending point; that is point B) to point B.1844

How are we going to get there?1852

Right here, we know that AD is parallel to BC; oh, that is written incorrectly, so let's fix that; AD is parallel to BC; they were both wrong.1855

AD is parallel to BC, and AE is congruent to CE.1888

We know that those are true, and then we are going to prove that this whole thing is a parallelogram.1897

In order to prove that this is a parallelogram, we have to think back to one of those theorems1906

and see which one we can use to prove that this is a parallelogram.1913

The first one that we can use is the definition of parallelogram.1917

If we can say that both pairs of opposite sides are parallel, then it is a parallelogram.1921

All we have is one pair; we don't know that this pair is parallel, or can we somehow say that it is parallel?1928

I don't think so; the only way that we can prove that these two are parallel is if we have an angle,1937

some kind of special angle relationship with transversals--like if I say that alternate interior angles are congruent,1946

same-side/consecutive interior angles are supplementary...if I say that corresponding angles are congruent...1957

if something, then the lines are parallel; I could do that.1964

For this one, it would be alternate interior angles--if they were congruent,1969

if it somehow gave me that, then I could say that these two lines are parallel, AB and DC.1973

And then, I could say that the whole thing is a parallelogram, because I have proved that it has two pairs of opposite sides being parallel.1980

But I can't do that, because I don't have that information.1989

Can I say that both pairs of opposite sides are congruent, from what is given to me? No.1994

Can I say that opposite angles are congruent? No.2002

I could say that these angles are congruent; they are vertical.2009

Or I could say that this angle and this angle are congruent, because they are vertical; but that is all I have with the angles.2013

Can I say that diagonals bisect each other?2020

Well, I have one diagonal that is bisected.2024

Can I somehow say that this diagonal is bisected?2027

I don't think so, just by being given parallel, congruent, and these angles--no.2034

Can I say that the last one works (remember the special theorem?)--one pair being both parallel and congruent?2040

We have that this pair is parallel; can we say that this pair is also congruent?2049

Well, because I can say that this angle is congruent to this angle...2056

let me do that in red; that way, you know that that is not the given...2067

since I know that these lines are parallel, if this acts as my transversal, I can say that this angle is congruent to this angle.2072

Remember: it is just line, line, transversal; angle, angle; do you see that?2091

This is B; this is D; this is this angle right here; and this is this angle right here.2104

I can say that those angles are congruent, because the lines are parallel.2114

Well, I can now prove that these two triangles are congruent, because of Side-Angle-Angle, or Angle-Angle-Side.2120

Therefore, the triangles are congruent; and then, these sides will be congruent, because of CPCTC.2130

And then, I can say that it is parallelogram, because of that theorem of one pair being both parallel and congruent.2140

Let me just explain that again, one more time.2149

I need to prove that this is a parallelogram with the information that is given to me.2152

All I have that is given is that this side and this side are parallel, and this and this are congruent.2157

From what is given to me, I can say that these angles are congruent, because they are vertical;2167

and I can say that these angles are congruent, because alternate interior angles are congruent when the lines are parallel.2174

The whole point of me doing all of this is to show this using a theorem that says, if one pair of opposite sides2182

are both parallel and congruent, then it is a parallelogram.2194

I want to show that this side is both parallel (which is given) and congruent, so that I can say that this whole thing is a parallelogram.2200

But the only way to show that this side is congruent is to prove that this triangle and this triangle are congruent,2209

so that these sides of the triangle will be congruent, based on CPCTC.2223

If you are still a little confused--you are still a little lost--then just follow my steps of my proof.2231

And then, hopefully, you will be able to see, step-by-step, what we are trying to do.2236

Step 1: my statements and my reasons (just right here): #1: the statement is the given,2241

AD is parallel to BC, and AE is congruent to CE; what is my reason? "Given"--it was given to me.2258

Then, my next step: I am going to say...2277

Now, the angles that are in red--that is not the given statement; it is not anything that is given, so I have to state it and list it out.2281

I am going to say, "Angle AED" (I can't say angle E, because see how angle E can be any one of these;2291

so I have to say angle AED) "is congruent to angle CEB"; what is the reason for that?--"vertical angles are congruent."2302

My #3: Angle ADE is congruent to angle CBE; what is the reason for that?2324

"If parallel lines are cut by a transversal," (now again, you can write it all out, or actually,2353

we could probably just say "alternate interior angles theorem"; or if your book doesn't have a name for that,2370

then you can just write it out) "then alternate interior angles are congruent."2379

Now, I am just writing it out for those of you that don't have the name for it.2390

If you do, then you can just go ahead and write that out, and that would just be "alternate interior angles theorem."2392

The fourth step: now that we said that we have this side with this side from the triangle,2400

AE inside CE (that is the side), then we have this angle with this angle; there is an angle;2408

and this angle with this angle--these are all corresponding parts of the two triangles.2419

Now, we can say that the two triangles...triangle AED is congruent to triangle CEB.2427

And that means that this whole triangle, now, is congruent to this whole triangle.2442

What is the reason? Angle-Angle-Side.2446

If you are unsure what this is, then go back to the section on proving triangles congruent.2451

And then, we just proved that these two triangles are congruent by Angle-Angle-Side--that reason.2459

And then, now that the triangles are congruent, we can say that any corresponding parts of the two triangles are congruent.2467

Now I can say that AD is congruent to CB, and the reason is CPCTC; and that is "Corresponding Parts of Congruent Triangles are Congruent."2477

Corresponding parts of the congruent triangles are going to be congruent.2494

So now that we have stated that those two sides are congruent, now we can go ahead and say that quadrilateral ABCD is a parallelogram.2500

And the reason would be "if one pair is both parallel and congruent, then it is a parallelogram."2515

Remember: this is point A and point B; this is the starting point and ending point.2539

So, see how I have this statement right here, and my last statement.2543

All we did was prove that these two sides are congruent, so that we could use the theorem that we just went over.2551

That is it for this lesson; thank you for watching Educator.com.2560