### Parallelograms

- Quadrilaterals are four-sided polygons
- Parallelogram: A quadrilateral with two pairs of parallel sides
- Properties of Parallelograms:
- Opposite sides of a parallelogram are congruent
- Opposite angles of a parallelogram are congruent
- Consecutive angles in a parallelogram are supplementary
- The diagonals of a parallelogram bisect each other

### Parallelograms

ABCD, BCDA, ACBD and ADBC

Consecutive angles in a parallelogram are congruent.

Oppostite angles in a parallelogram are congruent.

- m∠BDC = m∠ABD
- 2x + 5 = 8

- m∠ABD = 180 − m∠BAE − m∠AEB
- m∠ABD = 180 − 45 − 55 = 80
- m∠BDC = m∠ABD

- A( − 2, 1), B( − 3, − 3), C(3, − 2), D(4, 2).
- AB = √{( − 3 − ( − 2))
^{2}+ ( − 3 − 1)^{2}} = √{1 + 16} = √{17} - BC = √{(3 − ( − 3))
^{2}+ ( − 2 − ( − 3))^{2}} = √{36 + 1} = √{37} - DC = √{(3 − 4)
^{2}+ ( − 2 − 2)^{2}} = √{1 + 16} = √{17} - AD = √{(4 − ( − 2))
^{2}+ (2 − 1)^{2}} = √{36 + 1} = √{37} - AB = CD, AD = BC
- ―AB ≅ ―CD ,―AD ≅ ―BC

^{2}+ ( − 3 − 1)

^{2}} = √{1 + 16} = √{17}

BC = √{(3 − ( − 3))

^{2}+ ( − 2 − ( − 3))

^{2}} = √{36 + 1} = √{37}

DC = √{(3 − 4)

^{2}+ ( − 2 − 2)

^{2}} = √{1 + 16} = √{17}

AD = √{(4 − ( − 2))

^{2}+ (2 − 1)

^{2}} = √{36 + 1} = √{37}

AB = CD, AD = BC ―AB ≅ ―CD ,―AD ≅ ―BC

Quadrilatral ABCD is parallelogram.

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Parallelograms

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro
- Quadrilaterals
- Parallelograms
- Properties of Parallelograms
- Angles and Diagonals
- Consecutive Angles in a Parallelogram are Supplementary
- The Diagonals of a Parallelogram Bisect Each Other
- Extra Example 1: Complete Each Statement About the Parallelogram
- Extra Example 2: Find the Values of x, y, and z of the Parallelogram
- Extra Example 3: Find the Distance of Each Side to Verify the Parallelogram
- Extra Example 4: Slope of Parallelogram

- Intro 0:00
- Quadrilaterals 0:06
- Four-sided Polygons
- Non Examples of Quadrilaterals
- Parallelograms 1:35
- Parallelograms
- Properties of Parallelograms 4:28
- Opposite Sides of a Parallelogram are Congruent
- Opposite Angles of a Parallelogram are Congruent
- Angles and Diagonals 6:24
- Consecutive Angles in a Parallelogram are Supplementary
- The Diagonals of a Parallelogram Bisect Each Other
- Extra Example 1: Complete Each Statement About the Parallelogram 10:26
- Extra Example 2: Find the Values of x, y, and z of the Parallelogram 13:21
- Extra Example 3: Find the Distance of Each Side to Verify the Parallelogram 16:35
- Extra Example 4: Slope of Parallelogram 23:15

### Geometry Online Course

### Transcription: Parallelograms

*Welcome back to Educator.com.*0000

*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 name*0460

*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 (x _{1} - x_{2})...and this doesn't mean x squared,*1068

*or anything like that; it is just the first x, minus the second x, squared; plus y _{1}, 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

*x _{1} (it doesn't matter which one you use) is -1, minus -3, squared, plus y_{1}, 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 y _{2} - y_{1}, 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

1 answer

Last reply by: patrick guerin

Thu Sep 25, 2014 6:19 AM

Post by Aurelio Da Costa on September 9, 2013

I cant find the practice questions. Can anyone tell me how? Thanks heaps - Aurelio