### Rectangles

- Rectangle: Quadrilateral with four right angles
- Diagonals of Rectangles:
- If a parallelogram is a rectangle, then its diagonals are congruent
- If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle
- Rectangles Summary
- Opposite sides are congruent and parallel
- Opposite angles are congruent
- Consecutive angles are supplementary
- Diagonals are congruent and bisect each other
- All four angles are right angles

### Rectangles

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

Write all the congruent segments in rectangle ABCD.

Write all the right angles in rectangle ABCD.

If ―AC ≅ ―BD and ―AC bisect ―BD at E, then quadrilateral ABCD is a rectangle.

If quadrilateral ABCD is a rectangle, then ∆ ABC ≅ ∆ DCB.

m∠CAD = 30

^{o}, find m ACD.

- m∠CAD + m∠ACD + m∠ADC = 180
^{o} - m∠ACD = 180
^{o}− m∠CAD − m∠ADC - m∠ACD = 180
^{o}− 30^{o}− 90^{o}

^{o}.

If the oppsite sides of a quadrilateral are congruent and parallel, then the quadrilatetral is a rectangle.

BE = 3x + 9, AE = 2x + 6, find x.

- ―BE ≅ ―AE
- 3x + 9 = 2x + 6

AB = 4x + 5, CD = 17, find x.

- ―AB ≅ ―CD
- 4x + 5 = 17

- A( − 2, 1), B( − 1, − 3), C(3, − 2), D(2, 2)
- the slope is m
_{AB}= [( − 3 − 1)/( − 1 − ( − 2))] = − 4 - m
_{CD}= [(2 − ( − 2))/(2 − 3)] = − 4 - m
_{AD}= [(2 − 1)/(2 − ( − 2))] = [1/4] - m
_{BC}= [( − 2 − ( − 3))/(3 − ( − 1))] = [1/4] - m
_{AB}= m_{CD}, m_{AD}= m_{BC} - Quadrilateral ABCD is a parallelogram.
- AC = √{(3 − ( − 2))
^{2}+ ( − 2 − 1)^{2}} = √{25 + 9} = √{34} - BD = √{(2 − ( − 1))
^{2}+ (2 − ( − 3))^{2}} = √{9 + 25} = √{34} - AC = BD
- ―AC ≅ ―BD

*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

### Rectangles

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
- Rectangles
- Diagonals of Rectangles
- Proving a Rectangle
- Rectangles Summary
- Opposite Sides are Congruent and Parallel
- Opposite Angles are Congruent
- Consecutive Angles are Supplementary
- Diagonals are Congruent and Bisect Each Other
- All Four Angles are Right Angles
- Extra Example 1: Find the Value of x
- Extra Example 2: Name All Congruent Sides and Angles
- Extra Example 3: Always, Sometimes, or Never True
- Extra Example 4: Determine if ABCD is a Rectangle

- Intro 0:00
- Rectangles 0:03
- Definition of Rectangles
- Diagonals of Rectangles 2:52
- Rectangles: Diagonals Property 1
- Rectangles: Diagonals Property 2
- Proving a Rectangle 4:40
- Example: Determine Whether Parallelogram ABCD is a Rectangle
- Rectangles Summary 9:22
- Opposite Sides are Congruent and Parallel
- Opposite Angles are Congruent
- Consecutive Angles are Supplementary
- Diagonals are Congruent and Bisect Each Other
- All Four Angles are Right Angles
- Extra Example 1: Find the Value of x 11:03
- Extra Example 2: Name All Congruent Sides and Angles 13:52
- Extra Example 3: Always, Sometimes, or Never True 19:39
- Extra Example 4: Determine if ABCD is a Rectangle 26:45

### Geometry Online Course

### Transcription: Rectangles

*Welcome back to Educator.com.*0000

*The next lesson is on rectangles.*0001

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

*Now, rectangles are a special type of parallelogram; if I have a quadrilateral, we know that a quadrilateral is just a polygon*0017

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

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

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

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

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

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

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

*Those are all of the properties of a parallelogram.*0090

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

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

*we know that opposite angles are congruent; so this angle and this angle--in this case, all angles would be congruent*0115

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

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

*And we know that consecutive angles (the measure of angle B plus the measure of angle C) are 180.*0149

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

*Determine whether the parallelogram is a rectangle.*0291

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

*Let's count up from here to here: to count the slope, you are going to go 1, 2, 3; that is a positive 3; you went up positive 3.*0437

*And then, we are going to run 1, 2, 3, 4; so the slope of AB is positive 3 over 4.*0449

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

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

*1, 2--positive 2; so this one was 3/4, and this one is -3/2.*0481

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

*10x + 5, plus 55, equal to 90 degrees; so here, this is going to be 10x + 60 = 90.*0698

*Subtract the 60, so 10x = 30; and x = 3.*0716

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

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

*and plug it back into this right here, so that we would find the measure of angle 1.*0745

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

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

*Now, the next one: here is the diagonal; AC is 52, and DB (which is the other diagonal) is 5x + 2.*0766

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

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

*52 = 5x + 2; I am going to subtract the two, so I get 50 = 5x...divide the 5, and so, 10 is x, or x = 10.*0798

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

*In the same way, 1 is congruent to angle 7, and since these are congruent, then I know*1095

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

*This is the same type of concept.*1303

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

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

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

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

*So then, this is "always."*1333

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

*That is actually going to be "sometimes."*1452

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

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

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

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

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

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

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

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

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

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

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

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

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

*Then, either way, these both are true for #4.*1571

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

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

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

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

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

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

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

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

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

*So, slope is the difference of the y's, y _{2} - y_{1}, over the difference of the x's.*1666

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

*The difference of their y's would be 6 - 5, over -2 - 3, which is going to be 1 over...this is -5.*1680

*Then, I am going to find the slope of BC: 0 - 6, over 2 - -2.*1696

*This is going to be -6 over...this is 4; and then, that is going to be -3/2.*1713

*Now, AB has a slope of -1/5, and BC has a slope of -3/2.*1727

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

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

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

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

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

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

*Thank you for watching Educator.com.*1786

0 answers

Post by Tami Cummins on July 31, 2013

I meant isn't even a parallelogram is it?

3 answers

Last reply by: Professor Pyo

Thu Jan 2, 2014 3:52 PM

Post by Tami Cummins on July 31, 2013

On example 3 number 3. I thought the definition of parallelogram was that opposite sides are parallel and congruent. If that is true then wouldn't a parallelogram with a right angle have to have 4 right angles and thus be a rectangle. The example you gave is even a parallelogram is it?