WEBVTT mathematics/geometry/pyo
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Welcome back to Educator.com.
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The next lesson is on rectangles.
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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.
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Now, rectangles are a special type of parallelogram; if I have a quadrilateral, we know that a quadrilateral is just a polygon
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with four sides--any polygon with four sides is considered a quadrilateral.
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Then, a special type of quadrilateral is a parallelogram; and then, a special type of parallelogram is a rectangle.
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That means that a rectangle has all of the properties of a quadrilateral and a parallelogram.
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Now, a quadrilateral doesn't really have any properties, except that it just has four sides.
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For a parallelogram, though, we have a few: just to review: for a parallelogram, we know that opposite sides are parallel.
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We know that two pairs of opposite sides are congruent; we know that two pairs of opposite angles are congruent.
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We know that diagonals bisect each other; and we know that consecutive angles are supplementary.
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Those are all of the properties of a parallelogram.
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Now, since a rectangle is a special type of parallelogram, all of those properties of parallelograms apply to rectangles,
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which means that rectangles have opposite sides parallel and congruent (this is congruent also);
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we know that opposite angles are congruent; so this angle and this angle--in this case, all angles would be congruent
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to each other, because right angles are all congruent; we know that rectangles' diagonals bisect each other,
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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.
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And we know that consecutive angles (the measure of angle B plus the measure of angle C) are 180.
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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.
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All of the properties of a parallelogram apply to the rectangle.
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The diagonals of rectangles: if a parallelogram is a rectangle, then its diagonals are congruent.
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If we have a rectangle, this diagonal and this diagonal are going to be congruent.
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That means that the distance from here to here is the same as the distance from here to here.
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So then, this diagonal and this diagonal are congruent.
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If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.
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We know that a parallelogram's diagonals are not congruent; they could be, but they are not always.
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So, that wouldn't be considered a property of a parallelogram.
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The property of a parallelogram on diagonals is just that they bisect each other.
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And these congruent diagonals bisect each other, too; but they bisect each other, and they are congruent.
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It is like an additional property added on when you have rectangles.
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It is that all of the properties of a parallelogram are true, plus more.
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If the diagonals of a parallelogram are congruent, then it is a rectangle.
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These are the same theorem; it is just that this is the converse, because it is saying that,
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if it is a rectangle, then we know that the diagonals are congruent.
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But also, if the diagonals are congruent, then it is a rectangle; so it works vice versa--it works both ways.
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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.
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Determine whether the parallelogram is a rectangle.
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We know that these are parallel, because the slope of AB and the slope of DC are going to be the same,
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and the slope of AD and the slope of BC are the same.
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So, we know that it is a parallelogram; and automatically, we can assume that, even if we don't know what the slopes are,
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we can assume that these opposite lines are going to have the same slope, because it tells us that it is a parallelogram.
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So, now, what I want to know is if it is going to be a rectangle.
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How do we know that this would be a rectangle?
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We can still use slope; since this is already a parallelogram, we don't have to show their slopes--
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well, we do, but we don't have to show that opposite slopes are the same,
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because again, they tell us that it is a parallelogram.
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But what we do have to do is show that, since we know that slopes that are perpendicular have negative reciprocals,
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I know that this side, AB, and side BC are going to be perpendicular, because it is right angles.
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So, since these are right angles, in order for that to be a rectangle, their slopes have to be negative reciprocals of each other.
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I am going to find their slopes: let's see, the slope of AB is going to be, remember, rise over run.
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Slope is rise over run, how many you go up and down versus how many you go left and right.
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Remember: if you count up, that is a positive number; if you count down, that is a negative number.
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If you count right, that is a positive; and if you count left, then that is a negative.
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And we know that because, on the y-axis, as you go up, the numbers get bigger; they become more positive.
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If we go to the right, the same thing happens: the numbers go towards the positive numbers.
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If you go down, those are the negative numbers--you are going towards negatives.
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And then, if you go to the left, then the numbers are getting smaller again, to the negatives, so it is a negative number.
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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.
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And then, we are going to run 1, 2, 3, 4; so the slope of AB is positive 3 over 4.
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And then, let's see, the slope of BC: again, we have to go up and down first.
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I am going to go down: 1, 2, 3; I went down 3--that is negative 3, over...
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1, 2--positive 2; so this one was 3/4, and this one is -3/2.
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Now, they are not negative reciprocals of each other; this one is positive, and this one is negative,
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but then the negative reciprocal would be -4/3, but it is not.
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So, automatically, I don't have to go on anymore; I just know that these two are not perpendicular.
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Therefore, this is not a right angle, which means that this is not a rectangle.
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Now, if these two were negative reciprocals of each other, then I would have to continue and find the slope of DC,
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and then find the slope of AD, and make sure that they are also reciprocals of each other,
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because this could be a right angle, but then, this might not, or the other ones might not.
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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,
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then it will be the same for all, because it is a parallelogram, so opposite angles would have to be congruent.
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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."
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Opposite sides are congruent and parallel; those are both properties of a parallelogram.
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Opposite angles are congruent; that is also a property of a parallelogram.
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Consecutive angles are supplementary; that is a property of a parallelogram.
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Diagonals are congruent and bisect each other: well, for the property of a parallelogram, it is just that they bisect each other.
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For parallelograms, they are not always going to be congruent; so only these are the ones that are parallelograms:
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properties of parallelograms--that is the symbol for a parallelogram.
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Now, then new ones, the ones that are more specific to rectangles: diagonals are congruent--that is this one.
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And all four angles are right angles--there is another one that is just considered a property of a rectangle.
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All of the ones in red are properties of rectangles.
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Let's work on our examples: the first one: we are going to find the value of x.
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The measure of angle x here is 10x + 5, right here; and the measure of angle 2 is 55 degrees.
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Now, since these two angles, we know, are what?--they add up to 90--they are complementary, I just have to make those two,
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10x + 5, plus 55, equal to 90 degrees; so here, this is going to be 10x + 60 = 90.
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Subtract the 60, so 10x = 30; and x = 3.
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Oh, and then, just to look back, they were asking us for the value of x.
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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,
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and plug it back into this right here, so that we would find the measure of angle 1.
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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.
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But they are only asking us for the value of x, so that would be the answer.
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Now, the next one: here is the diagonal; AC is 52, and DB (which is the other diagonal) is 5x + 2.
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Now, if you remember, the property of a rectangle is that diagonals are congruent.
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So then, we know that AC is congruent to DB; that means that I can just make them equal to each other.
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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.
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Both of these use the properties of the rectangle that are more specific to the rectangle,
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where each angle of a rectangle is a right angle, and the diagonals are congruent.
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The next example: Name all congruent sides and angles.
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All we have to do is just name all of the sides that are congruent and all of the angles that are congruent.
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First, I know that AB is congruent to CD (we are doing sides first), and then AD is congruent to CB.
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And then, the diagonals (because this is a rectangle): DB is congruent to CA.
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And now for the angles: this is a little bit different, because...well, we know that all of the angles are congruent.
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Angle A is congruent to angle B, which is congruent to angle C, and it is also congruent to angle D; why?
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Because all right angles are congruent; we know that each of these is a right angle.
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But then, we have these diagonals written here; so that means that all of the angles are split up now.
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So, we have to see what angles are congruent here.
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Since we know that these sides are parallel (that is a property of a parallelogram, and all properties of parallelograms apply to rectangles),
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angle 1 and angle 7 are alternate interior angles; so if I were to draw that again, here is, let's say, AB extended.
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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.
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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.
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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,
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all of these are actually equal parts; so if you can see that this is a triangle (an isosceles triangle, to be more specific),
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because this side and this side are congruent, then we know that these angles are congruent,
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because of the base angles theorem, or you can say the isosceles triangle theorem.
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I know that angles 2 and 5 are congruent, because of that reason there.
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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,
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which means that (I am trying to get that red)...all of those would be congruent.
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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.
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In the same way, 1 is congruent to angle 7, and since these are congruent, then I know
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that 6 and 7 are congruent, because of the base angles theorem of this triangle.
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And then, this is alternate interior angles with angle 3, so all four of those angles are congruent.
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Now, we also have these four angles in the middle right here.
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Now, I can't say for sure if these angles are going to be congruent to any of the outer angles,
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because we just can't say; we don't know what the measures of those angles are.
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But what I can say is that angle 9 is congruent to angle 12, because they are vertical angles.
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And then, angle 10 is congruent to angle 11, because they are vertical, as well.
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So, there is a lot of stuff right there: all of the congruent sides and the congruent angles.
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For the next example, we are going to determine whether each statement is always true, sometimes true, or never true.
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If a quadrilateral is a rectangle, then it is a parallelogram.
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Now, if I say "quadrilateral," that is a very general name for a polygon.
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And then, it gets more specific to a parallelogram, and then it gives another more specific name, a type of parallelogram, to a rectangle.
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This is showing you where it starts from: a quadrilateral is the big picture; it is what the general polygon is called.
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Then, it is down to this type; if it is like this, then it is a parallelogram.
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And then, if a parallelogram is like this, then it is a rectangle.
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In the same way, I can maybe use, let's say, dogs.
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Now, if I say "animals," saying "animal" is like saying "quadrilateral"; it is very general; it is just very broad.
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And then, a type of animal would be, let's say, "dogs."
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And then, a more specific type would be, let's say, a Chihuahua.
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This is the same type of concept.
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Now, using that, let's look at these problems again: if a quadrilateral is a rectangle, then it is a parallelogram.
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So, if an animal is a Chihuahua, then that Chihuahua is a dog.
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Is that true? Is that always true, sometimes true, or never true?
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This would be always; if it is a Chihuahua, then it is always a dog; Chihuahuas are always dogs.
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So then, this is "always."
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If a quadrilateral is a parallelogram, then it is a rectangle; now, are parallelograms always rectangles?
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No, sometimes they are just parallelograms; in the same way, if the animal was a dog, then it is a Chihuahua.
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Well, dogs can be something else; there are different types of dogs.
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There is the golden retriever; there is the Maltese; there are all of these different dogs.
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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.
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But it can be; so then, this one would be "sometimes."
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If a parallelogram has a right angle, then it is a rectangle.
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If a parallelogram has a right angle, then automatically, a right angle is a property of a rectangle.
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Now, even though rectangles say that there are four right angles, just the fact that there is one in a parallelogram--
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that means that all four have to be right angles, because from this, we know that if there is just one,
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well, opposite angles are congruent; that means that this one has to be one, too.
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Well, aren't consecutive angles supplementary?--so, if this is a right angle, if this is 90 degrees,
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180 - 90 is 90, so this has to be 90; and then, this is also the same as that, so...
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If there is one right angle in a parallelogram, then it is a rectangle--then all four would be right angles.
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Now, if they said, "If a quadrilateral has a right angle, then it is a rectangle," that is not going to be "always."
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That is actually going to be "sometimes."
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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.
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So, this is a quadrilateral, and this is a parallelogram.
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A parallelogram with a right angle would make it a rectangle.
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A quadrilateral with a right angle is not always going to be a rectangle.
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Then, this would be "always," because they said "parallelogram."
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The last one: If opposite angles of a quadrilateral are congruent, then it is a rectangle.
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Now, notice how they say the word "quadrilateral"; they don't say "parallelogram."
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Can you draw a counter-example? Remember: this is a counter-example; it is an example showing the opposite.
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If opposite angles of a quadrilateral are congruent, can you draw a quadrilateral with opposite angles being congruent, but it is not a rectangle?
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Yes, I can; how about that? Opposite angles are congruent, and that is a quadrilateral.
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So, opposite angles of a quadrilateral are congruent, but it isn't a rectangle.
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It could be, because I could also draw something like this, where opposite angles are congruent, which would just be like this.
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Or if you want, just show that they are congruent and opposite.
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Then, either way, these both are true for #4.
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This would be "sometimes," meaning that opposite angles of a quadrilateral can be congruent, but it doesn't always have to be a rectangle.
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Sometimes, it could be, in this case; and it is not in this case; so it is sometimes.
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And the last one, the fourth example: We are going to determine if ABCD is a rectangle, given all of their vertices.
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Again, we want to find their slope, because we know that rectangles have perpendicular sides.
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We know that perpendicular lines have negative reciprocals of each other--their slopes are negative reciprocals of each other.
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So, we are going to find the slopes of each of the lines, and then see if they are negative reciprocals.
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Just to draw a little rectangle: this is A, B, C, and D.
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You know what sides have to be perpendicular with what other sides.
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The slope of, let's say first, AB: now, if you want to do this problem, you have to know the formula for slope.
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So, slope is the difference of the y's, y₂ - y₁, over the difference of the x's.
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Here we are going to use this point and this point.
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The difference of their y's would be 6 - 5, over -2 - 3, which is going to be 1 over...this is -5.
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Then, I am going to find the slope of BC: 0 - 6, over 2 - -2.
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This is going to be -6 over...this is 4; and then, that is going to be -3/2.
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Now, AB has a slope of -1/5, and BC has a slope of -3/2.
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So, they are not perpendicular, because their slopes are very different; they are not negative inverses, or reciprocals, of each other.
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Therefore, this is not a rectangle; so then, this is "no--not a rectangle."
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That is it for this lesson; we are going to cover other types of parallelograms.
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We are going to go over the square and the rhombus.
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And then, after that, we are going to go over the trapezoid and kites.
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So, we are going to go over different types of parallelograms next.
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Thank you for watching Educator.com.