### Proving Parallelograms

- Parallelogram Theorems:
- If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram
- If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram
- If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram
- If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram
- A quadrilateral is a parallelogram if:
- Both pairs of opposite sides are parallel
- Both pairs of opposite sides are congruent
- Both pairs of opposite angles are congruent
- Diagonals bisect each other
- A pair of opposite sides is both parallel and congruent

### Proving Parallelograms

If ―AB ≅ ―CD and ―AD ≅ ―BC , then quadrilateral ABCD is a parallelogram.

If both sides of opposite sides of a quadrilateral are parallel, then it is _____ a parallelogram.

If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

If four angles of a quadrilateral are all congruent to each other, then the quadrilateral is a parallelogram.

―AE = 15, ―AD = 25, find the measurement of ―AC and ―BC .

- ―AC = 2―AE
- ―AC = 30
- ―BC = ―AD

Parallelogram ABCD, AB = 4x − 6, CD = 3x + 8, find x.

- ―AB ≅ ―CD
- 4x − 6 = 3x + 8

Parallelogram ABCD, m∠B = 2x + 10, m∠A = 28, find x.

- m∠A + m∠B = 180
- 28 + 2x + 10 = 180
- 2x = 142

A(3, − 2), B(2, 3), C( − 4, 4), D( − 3, − 3).

- AB = √{(2 − 3)
^{2}+ (3 − ( − 2))^{2}} = √{1 + 25} = √{26} - CD = √{( − 3 − ( − 4))
^{2}+ ( − 3 − 4)^{2}} = √{1 + 49} = √{50} - AB ≠ CD

Determine whether the following statement is true or false.

If quadrilateral ABCD is a parallelogram, then ∆ ABE ≅ ∆ CDE.

Given: ―AD ||―BC , ∠BAC ≅ ∠DCA

Prove: Quadrilateral ABCD is a parallelogram.

- Statements; Reasons
- ―AD ||―BC; Given
- ∠DAC ≅ ∠BCA ; Alternate interior angles
- ∠BAC ≅ ∠DCA ; Given
- ―AC ≅ ―AC ; Reflexive prop ( = )
- ∆ BAC ≅ ∆ DCA ; ASA
- ―AD ≅ ―BC ; Definition of congrent triangle
- ―AD ||―BC ; Given
- Quadrilateral ABCD is a parallelogram; Parallelograms theorem.

- S
- tatements; Reasons ―AD ||―BC; Given ∠DAC ≅ ∠BCA ; Alternate interior angles BAC ≅ ∠DCA ; Given ―AC ≅ ―AC ; Reflexive prop ( = ) ∆ BAC ≅ ∆ DCA ; ASA ―AD ≅ ―BC ; Definition of congrent triangle ―AD ||―BC ; Given Quadrilateral ABCD is a parallelogram; Parallelograms theorem

*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

### Proving 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
- Parallelogram Theorems
- Parallelogram Theorems, Cont.
- Proving Parallelogram
- Summary
- Both Pairs of Opposite Sides are Parallel
- Both Pairs of Opposite Sides are Congruent
- Both Pairs of Opposite Angles are Congruent
- Diagonals Bisect Each Other
- A Pair of Opposite Sides is Both Parallel and Congruent
- Extra Example 1: Determine if Each Quadrilateral is a Parallelogram
- Extra Example 2: Find the Value of x and y
- Extra Example 3: Determine if the Quadrilateral ABCD is a Parallelogram
- Extra Example 4: Two-column Proof

- Intro 0:00
- Parallelogram Theorems 0:09
- Theorem 1
- Theorem 2
- Parallelogram Theorems, Cont. 3:10
- Theorem 3
- Theorem 4
- Proving Parallelogram 6:21
- Example: Determine if Quadrilateral ABCD is a Parallelogram
- Summary 14:01
- Both Pairs of Opposite Sides are Parallel
- Both Pairs of Opposite Sides are Congruent
- Both Pairs of Opposite Angles are Congruent
- Diagonals Bisect Each Other
- A Pair of Opposite Sides is Both Parallel and Congruent
- Extra Example 1: Determine if Each Quadrilateral is a Parallelogram 16:54
- Extra Example 2: Find the Value of x and y 20:23
- Extra Example 3: Determine if the Quadrilateral ABCD is a Parallelogram 24:05
- Extra Example 4: Two-column Proof 30:28

### Geometry Online Course

### Transcription: 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 of*0541

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

*Slope is y _{2} - y_{1}, over x_{2} - x_{1}, 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 -4*0692

*(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 show*0909

*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 theorem*0966

*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 (x _{1} - x_{2})^{2} + (y_{1} - y_{2})^{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 A*1840

*(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 theorems*1906

*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 sides*2182

*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

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