WEBVTT mathematics/geometry/pyo
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Welcome back to Educator.com.
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For this lesson, we are going to use the theorems and the properties you learned in the previous lesson to prove parallelograms.
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Turning the properties that we learned into actual theorems, if/then statements:
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the first one: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
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Now, these theorems have no name; we have no name for the actual theorem, so we actually have to write it all out.
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If I say, "If opposite sides are congruent, then it is a parallelogram," you can shorten it in that way.
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So, if you ever have to use this theorem on a proof, then you can just shorten this as your reason,
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instead of having to write this whole thing out; "if opposite sides are congruent, then it is a parallelogram."
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Do something like that; you can just shorten words and phrases.
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Then, our conditional statement: as long as we have opposite sides being congruent...if this, then parallelogram.
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And this just means "parallelogram"; or actually, I can write it all out; maybe that will not be as confusing: "then parallelogram."
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The second one: "If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram."
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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.
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In the same way, the converse would be, "If both pairs of opposite angles are congruent, then it is a parallelogram."
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It is just basically saying that if the opposite angles are congruent, then it is a parallelogram.
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So, as long as we can prove this or this, then we can prove that it is a parallelogram.
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Now, we have other options, too; there are actually more theorems.
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The third theorem that we can use to prove quadrilaterals parallelograms is on their diagonals.
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If we can prove that the diagonals (you can just say "if diagonals") bisect each other, then it is a parallelogram.
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You can shorten it in that way; if you can just prove that the diagonals bisect each other,
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in that way, then you have proven that the quadrilateral is a parallelogram.
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Oh, I had it right...parallelogram.
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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."
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This is one that wasn't on the previous lesson; this is actually not a property of a parallelogram.
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This is just an extra theorem that says that if you can prove that only one pair of opposite sides is both,
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parallel and congruent, then you can prove that it is a parallelogram.
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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.
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That is one way that you can prove that it is a parallelogram.
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With other theorems, you have to prove two pairs: the first one was two pairs of opposite sides being congruent;
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the second one was two pairs of opposite angles being congruent; for this one, you have to prove that both diagonals bisect each other.
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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.
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So then, you can just shorten it by saying, "If one pair of opposite sides is parallel and congruent, then it is a parallelogram."
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Maybe you can say something like that--just shorten it like that, in that way.
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This right here--we are just determining if this quadrilateral is a parallelogram.
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In the previous lesson, we did a couple of these; in that case, the problems before in the last lesson,
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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...
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For this problem, we have to determine if it is a parallelogram.
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We don't know that it is a parallelogram; so then, using the same methods, using the distance formula,
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we have to see if it is going to come out to be the same.
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If these two are the same, and these two are the same, then we have to say that it is a parallelogram.
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So, it is the same thing; you are using the same methods.
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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.
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And that is it--just verifying; you were just giving the measurements of them.
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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.
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Again, you can use the distance formula, or you can use slope.
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If you are going to use the distance formula to show that these opposite sides are congruent,
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and that these opposite sides are congruent, then you are going to be using the first theorem we went over,
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saying that if two pairs of opposite sides are congruent, then it is a parallelogram.
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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.
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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.
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I am not using one of the theorems, because remember: we said that if you state that two pairs of opposite sides are parallel,
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that is just the definition of a parallelogram; so by definition, we can say that it is a parallelogram, if we use slope,
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because then we are showing that opposite sides are parallel.
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We are not using one of the theorems; we are actually just using the definition of a parallelogram.
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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.
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And then, the distance formula, if you wanted to use that, is the square root of
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the first x minus the second x, squared, plus the first y minus the second y, squared.
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Slope is y₂ - y₁, over x₂ - x₁, or rise over run.
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Rise measures up/down; run measures left/right.
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In this case, slope will probably be a little bit easier, because for slope, all you have to do is count.
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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.
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This also: if you have the points written out for you, then this can be pretty easy.
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But we are just going to use the rise and run to find the slope by counting.
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When you move up, that is a positive number, and that is going to go on the top, in the numerator.
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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.
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So then, that is because when you go up, you are going towards the positive y-axis.
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If you to the right, you are going towards the positive x-axis.
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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.
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If you move left, you are going towards the negative x numbers, so that is also a negative number.
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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.
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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...
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we go to the right 1, so the slope is 3/1, or just 3. The slope of AB is 3.
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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.
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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
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(because going down is negative)...then from here, I am going to go 1, 2, 3, 4.
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So, I went to the right 4, and that is a positive, because I went to the right, which makes this slope -1.
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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,
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then I am going to count 1, 2, 3, down 3; so the slope of CD is down 3, which is -3, over...
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from here, I am going to go left 1; left 1 is -1; so then, -3/-1 is 3.
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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;
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then 1, 2, 3, 4...that is a negative 4; I am going to the left 4.
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And that makes this a negative 1; so since AB and CD have the same slope, I know that AB is parallel to CD.
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And BC and AD have the same slope; that means that they are also parallel.
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So, BC is also parallel to AD; I have two pairs of opposite sides parallel.
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So, by the definition of parallelogram, this is a parallelogram, so yes, quadrilateral ABCD is a parallelogram.
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OK, let's just summarize over the different theorems that we can use to prove parallelograms, before we actually start our examples.
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A quadrilateral is a parallelogram if any one of these is true.
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You don't have to prove all of these; just prove one of them.
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If you prove one of these, then you can prove that the quadrilateral is a parallelogram.
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The first one: a quadrilateral is a parallelogram if both pairs of opposite sides are parallel.
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That is the definition of parallelogram; so as long as you can prove (this is the definition of parallelogram)--
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as long as you can show--that this side is parallel to this side, and this side is parallel to this side,
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then by the definition of parallelogram, the quadrilateral is a parallelogram.
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The second one: If both pairs of opposite sides are congruent...as long as you show
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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.
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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.
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And remember: it has to be two pairs of opposite angles being congruent.
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Then, that is a parallelogram.
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Diagonals bisect each other--not "diagonals are congruent," but "they bisect each other."
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That means that this diagonal is cut in half, and this diagonal is cut in half.
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Those two halves are congruent; then this is a parallelogram.
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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
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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,
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as long as you can prove that that one pair of opposite sides) is both parallel and congruent, then this will be a parallelogram."
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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.
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Then, you can do that to prove parallelograms.
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Let's actually go through some examples now: the first one: Let's determine if each quadrilateral is a parallelogram.
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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.
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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.
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So, if I see that one pair of opposite sides is parallel, and the other side is congruent, that is not a parallelogram.
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This could be a parallelogram, but there is no theorem, and there is no definition, that says this.
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The closest one...well, there are a few; one of them says that it has to be both pairs of opposite sides being parallel.
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We have one pair being parallel; if these two sides were parallel, then we could use the definition of parallelogram.
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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.
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And then, the last one, the special one that we went over--that has to be the same pair.
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So, one pair, the same pair of opposite sides, being both parallel and congruent--then it is a parallelogram.
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So, if these sides are both parallel and congruent, then we have a parallelogram.
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Or these sides--if they were both parallel and congruent, then we can use that one; but it is none of those.
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So, this one is "no"; we cannot determine it.
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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."
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The next one: all I have are four right angles--nothing else; just four right angles.
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Now, for this one, the theorem that has to do with angles is "opposite angles are congruent."
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Now, this angle and this angle--are they congruent? Yes, they are.
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This angle and this angle--are they congruent? Yes, they are.
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So, this, therefore, is a parallelogram, so this one is "yes."
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We can use that one theorem that says that two pairs of opposite angles are congruent.
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Now, some of you are probably looking at this and thinking, "But that is a rectangle!"
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Yes, it is a rectangle; we are actually going to go over rectangles next lesson.
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But a rectangle is a special type of parallelogram, so rectangles are parallelograms.
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So, is this a parallelogram? Yes, it is.
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So, if we have a rectangle, then we have a parallelogram.
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But then, without even thinking of rectangles, with this alone, just looking at the angles, opposite angles are congruent;
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so we have two pairs of opposite angles being congruent.
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By that theorem, we have a parallelogram; so this is "yes."
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All right, the next one: Find the value of x and y to ensure that each is a parallelogram.
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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.
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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.
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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.
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Then, I divide the 4 from each side, and y is equal to 6.
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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.
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And then, for x, again, I have to do the same thing: so, 2x + 3 is equal to 3x - 4.
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I am going to subtract the 2x here; you can add the 4 to this, so 7 is equal to x.
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If x is 7, then this will be 14; this will be 17; here, this is 21 - 4, is 17.
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The next one: now, this looks like it would be a square or rectangle, but you can't assume that.
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I don't have anything that tells me that these are right angles; I don't have anything that tells me anything, really.
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I have to find x and y so that these diagonals will be bisected, because that is what I am working with.
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Then, this and this have to be congruent; this and this have to be congruent.
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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.
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Let's see, the y's: this y and then this...this one is y + 4 = 20 - 3y.
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So, if I add 3y, this is 4y + 4 = 20; subtract the 4; 4y = 16; divide the 4, and y = 4.
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So then, now, if x is 4, and y is 4, then these parts of the diagonals will be congruent.
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Therefore, the diagonals bisect each other, and then as a result, this is a parallelogram.
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The next example: Determine if the quadrilateral ABCD is a parallelogram.
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We are given the coordinates of all of the vertices of the quadrilateral.
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And then, we have to determine if it is a parallelogram.
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I can just draw it out here; it doesn't matter how you draw it, as long as, remember, when we label this out,
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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.
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It has to be in the order, consecutive.
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And I am drawing this just to show which coordinates are next to each other, which ones are consecutive.
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Again, you can use slope, or you can use the distance formula.
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Since we used slope last time, let's use the distance formula this time.
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I am going to find the distance of AB, compare that to the distance of CD, and see if they are congruent.
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And then, before you move on, why don't you just try those two and see if they are congruent,
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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."
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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.
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The distance formula is (x₁ - x₂)² + (y₁ - y₂)².
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The distance of AB is the square root of 5 - 9, squared, plus 6 - 0, squared.
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5 - 9 is -4, squared; plus 6 squared...this is 16 + 36, which is 52.
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Now, you can go ahead and simplify it if you want.
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Your teacher might want you to simplify it.
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But since all we are doing is just comparing to see if AB and CD are going to be the same,
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I can just leave it like that, and then see if CD is going to come out to be the same thing.
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If your teacher wants you to actually find the distance of each side and show the distance,
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and make it simplified or round it to the nearest decimal, then you have to simplify that.
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Or else, if it is just to determine if it is a parallelogram, then you can just leave it.
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A way to simplify that, though, just to show you: we know that 52 is not a perfect square.
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So, what you can do is a factor tree: 52...2 is a prime number, and 26; 2--circle it--and 13.
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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.
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So then, this is 2√13.
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CD next: CD is, using these two, 8 - 3 squared, plus -5 - 2, squared.
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Oh, -5 minus a -2...that is a plus.
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And then, the square root of...this is 5 squared, plus -3 squared; 25 + 9...this is 34.
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We found AB and CD, and they are not the same; let me just double-check.
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Let's double-check our work; this is 5 - 9, squared; 6 - 0, squared.
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And then, for CD, it is 8 - 3, squared, and -5 - -2, squared.
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We have 16 + 36, which is 52, so √52; the square root of 5 squared plus -3 squared is 25 + 9, which is √34.
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So, I know that, since these are not congruent (this is √52, and this is √34), they are different.
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I can stop here; I don't have to continue and show my other two sides (again, unless your teacher wants you to).
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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."
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No, quadrilateral ABCD is not a parallelogram, because opposite sides are not congruent.
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If it was congruent, if they were the same, then you would have to go ahead and find the distance of BC,
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find the distance of AD, and then compare those two.
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For the last example, we are going to complete a proof of showing that it is a parallelogram.
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Always look at your given; using your given, you are going to go from point A
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(this is your point A; this is your starting point, and then this is your ending point; that is point B) to point B.
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How are we going to get there?
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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.
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AD is parallel to BC, and AE is congruent to CE.
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We know that those are true, and then we are going to prove that this whole thing is a parallelogram.
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In order to prove that this is a parallelogram, we have to think back to one of those theorems
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and see which one we can use to prove that this is a parallelogram.
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The first one that we can use is the definition of parallelogram.
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If we can say that both pairs of opposite sides are parallel, then it is a parallelogram.
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All we have is one pair; we don't know that this pair is parallel, or can we somehow say that it is parallel?
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I don't think so; the only way that we can prove that these two are parallel is if we have an angle,
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some kind of special angle relationship with transversals--like if I say that alternate interior angles are congruent,
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same-side/consecutive interior angles are supplementary...if I say that corresponding angles are congruent...
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if something, then the lines are parallel; I could do that.
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For this one, it would be alternate interior angles--if they were congruent,
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if it somehow gave me that, then I could say that these two lines are parallel, AB and DC.
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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.
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But I can't do that, because I don't have that information.
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Can I say that both pairs of opposite sides are congruent, from what is given to me? No.
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Can I say that opposite angles are congruent? No.
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I could say that these angles are congruent; they are vertical.
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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.
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Can I say that diagonals bisect each other?
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Well, I have one diagonal that is bisected.
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Can I somehow say that this diagonal is bisected?
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I don't think so, just by being given parallel, congruent, and these angles--no.
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Can I say that the last one works (remember the special theorem?)--one pair being both parallel and congruent?
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We have that this pair is parallel; can we say that this pair is also congruent?
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Well, because I can say that this angle is congruent to this angle...
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let me do that in red; that way, you know that that is not the given...
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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.
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Remember: it is just line, line, transversal; angle, angle; do you see that?
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This is B; this is D; this is this angle right here; and this is this angle right here.
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I can say that those angles are congruent, because the lines are parallel.
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Well, I can now prove that these two triangles are congruent, because of Side-Angle-Angle, or Angle-Angle-Side.
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Therefore, the triangles are congruent; and then, these sides will be congruent, because of CPCTC.
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And then, I can say that it is parallelogram, because of that theorem of one pair being both parallel and congruent.
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Let me just explain that again, one more time.
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I need to prove that this is a parallelogram with the information that is given to me.
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All I have that is given is that this side and this side are parallel, and this and this are congruent.
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From what is given to me, I can say that these angles are congruent, because they are vertical;
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and I can say that these angles are congruent, because alternate interior angles are congruent when the lines are parallel.
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The whole point of me doing all of this is to show this using a theorem that says, if one pair of opposite sides
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are both parallel and congruent, then it is a parallelogram.
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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.
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But the only way to show that this side is congruent is to prove that this triangle and this triangle are congruent,
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so that these sides of the triangle will be congruent, based on CPCTC.
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If you are still a little confused--you are still a little lost--then just follow my steps of my proof.
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And then, hopefully, you will be able to see, step-by-step, what we are trying to do.
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Step 1: my statements and my reasons (just right here): #1: the statement is the given,
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AD is parallel to BC, and AE is congruent to CE; what is my reason? "Given"--it was given to me.
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Then, my next step: I am going to say...
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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.
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I am going to say, "Angle AED" (I can't say angle E, because see how angle E can be any one of these;
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so I have to say angle AED) "is congruent to angle CEB"; what is the reason for that?--"vertical angles are congruent."
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My #3: Angle ADE is congruent to angle CBE; what is the reason for that?
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"If parallel lines are cut by a transversal," (now again, you can write it all out, or actually,
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we could probably just say "alternate interior angles theorem"; or if your book doesn't have a name for that,
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then you can just write it out) "then alternate interior angles are congruent."
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Now, I am just writing it out for those of you that don't have the name for it.
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If you do, then you can just go ahead and write that out, and that would just be "alternate interior angles theorem."
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The fourth step: now that we said that we have this side with this side from the triangle,
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AE inside CE (that is the side), then we have this angle with this angle; there is an angle;
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and this angle with this angle--these are all corresponding parts of the two triangles.
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Now, we can say that the two triangles...triangle AED is congruent to triangle CEB.
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And that means that this whole triangle, now, is congruent to this whole triangle.
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What is the reason? Angle-Angle-Side.
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If you are unsure what this is, then go back to the section on proving triangles congruent.
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And then, we just proved that these two triangles are congruent by Angle-Angle-Side--that reason.
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And then, now that the triangles are congruent, we can say that any corresponding parts of the two triangles are congruent.
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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."
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Corresponding parts of the congruent triangles are going to be congruent.
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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.
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And the reason would be "if one pair is both parallel and congruent, then it is a parallelogram."
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Remember: this is point A and point B; this is the starting point and ending point.
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So, see how I have this statement right here, and my last statement.
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All we did was prove that these two sides are congruent, so that we could use the theorem that we just went over.
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That is it for this lesson; thank you for watching Educator.com.