WEBVTT mathematics/geometry/pyo
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Welcome back to Educator.com.
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The next lesson is on parallelograms; and before we talk about parallelograms, let's talk about quadrilaterals.
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A **quadrilateral** is a four-sided polygon; now, we know that a polygon is a shape that is closed off.
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It is a shape with three or more sides that is completely closed, and each side has to be a straight line.
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When we have a polygon that is four-sided, that is a quadrilateral.
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This is considered a quadrilateral; this is a quadrilateral; any type of shape where I have four sides is a quadrilateral.
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Now, some non-examples, some examples that are not quadrilaterals, would be maybe something like that,
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where it doesn't close; that is not a quadrilateral; or if I have something that overlaps like that,
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if two sides overlap, this is not a quadrilateral; and if I have something that maybe is curved like that,
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a side that is curved, that would not be a quadrilateral, as well.
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Again, a quadrilateral is a four-sided polygon, something that is four-sided.
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Each side has to be a straight line, and it is closed, and there are no overlapping sides.
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A **parallelogram** is a special type of quadrilateral, meaning that it is a four-sided polygon, but it has two pairs of parallel sides.
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That is why it is called a parallelogram, because there are two pairs of parallel sides.
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To show that my sides are parallel, I can do this; that means that those sides are parallel.
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To show that these sides are parallel, I am going to draw two of them.
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These two are parallel, and then these two are parallel; so we have two pairs of parallel sides.
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Now, if I label this as A, B, C, and D, then I can name this parallelogram by its symbol; that would be the symbol for a parallelogram.
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Parallelogram ABCD: that is how I can name it, just like when I have a triangle; I can say triangle ABC.
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In the same way, I can say that this is the parallelogram ABCD.
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I can also say parallelogram BCDA; I could say parallelogram CDAB, and so on, as long as the four vertices that I name are in order.
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So, I can say ADCB; I can say DABC; but I can't say parallelogram ACBD, because it skips.
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You are skipping a vertex; when you label your parallelogram, make sure that they are in order--you are calling it in order.
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It could be this way; it could be to the right, clockwise, or counterclockwise; it does not matter, as long as they are back-to-back.
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This is something that you cannot say.
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Now, when we talk about opposite pairs, we have two pairs of opposite sides and two pairs of opposite angles.
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I know that side AB and side DC are opposite sides, and angle A with angle C are opposite angles.
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And angle B with angle D are opposite angles.
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Some properties of parallelograms: now, these properties are in the form of theorems,
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just so you know that these are considered theorems.
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The first property is that opposite sides of a parallelogram are congruent.
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As long as we have a parallelogram, and you know that it is a parallelogram, they either tell you that it is a parallelogram,
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or if they show you these symbols; the definition of a parallelogram is a quadrilateral with two pairs of opposite sides that are parallel.
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That is the definition of a parallelogram; that is not a property--that is what it means to be a parallelogram, the definition.
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The properties are these, saying that, if this is a parallelogram, then opposite sides are congruent.
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That means that this side with this opposite side are congruent, and this side with this side are congruent.
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So then, if this is 5, then this will also be 5; if this is 8, then this is also 8.
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Opposite sides are congruent; there will be two pairs--there is one pair, and this is the other pair.
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Opposite angles of a parallelogram are congruent; that is the next property.
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That means that this angle and this angle are congruent, and that angle with this angle are congruent.
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So, as long as we have a parallelogram like that, then I know that opposite angles are congruent.
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So far, opposite sides are congruent, and opposite angles are congruent.
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Then, the next one: Consecutive angles in a parallelogram are supplementary.
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So then, I am just going to show that...I don't have to do this, because I am already telling you that this is a parallelogram;
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but just so you get used to seeing that opposite sides are parallel...
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Consecutive angles of a parallelogram: that means this angle, so if I have angle 1 and 2, they are going to be supplementary.
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And one way I can show you why this is true, or how this is true: if I extend this side out, and extend this side out,
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then extend this side, then...now, I know that these two sides are parallel; this is like a transversal.
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If you remember that one section, that chapter that we went over, about parallel lines being cut by a transversal,
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then remember that we had special angle relationships; so remember how this angle right here,
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and (I am actually going to call that angle 2, just so that it is the same as this diagram right here)
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this is angle 1, and this is angle 2; now, we know that the name
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for this angle relationship is consecutive interior angles, or same-side interior angles.
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And we know that they are supplementary; as long as the lines are parallel, then these angles are supplementary.
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So, it is the same thing here: these are consecutive interior angles.
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Then these are supplementary; they are all the consecutive angles in a parallelogram.
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So, if I know that these two lines are parallel, this is a transversal;
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then, my interior angles are supplementary, in the same way that consecutive interior angles are supplementary.
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Then, these two are supplementary; this angle with this angle is supplementary;
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this angle with this angle is supplementary; and this angle with this angle is supplementary.
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Opposite angles are congruent, but consecutive angles are supplementary.
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Diagonals of a parallelogram bisect each other.
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If I draw diagonals, those are my diagonals; now, they are not congruent--diagonals are not congruent.
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That is a common mistake made by students--saying that these diagonals are congruent.
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Diagonals of a parallelogram are not congruent; obviously, it looks like this...from here...
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if you were to walk from this point to this point, that looks further than this point to this point.
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So, you can't say that this diagonal is the same as this diagonal.
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Diagonals are not congruent; they actually just bisect each other--that means that this diagonal cuts this diagonal in half;
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this got cut in half; and then, this diagonal cuts this diagonal in half, so that means these are congruent.
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The two diagonals are not congruent to each other; but the two halves of each diagonal are congruent.
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So then, only this part and this part of the same diagonal are congruent; this part and this part of the same diagonal are congruent.
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Another mistake is saying that this part, this half, of this diagonal is congruent to this half of the other diagonal.
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No; only the diagonal is cut in half--each diagonal is just cut in half; that is all you have to think of.
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They just bisect each other, and that is it.
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We went over four properties so far, before we start our next example.
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The first property of parallelograms...
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Well, first of all, the definition of parallelogram says that we have a quadrilateral (meaning that it is a four-sided polygon)
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with two pairs of opposite sides parallel; that is a parallelogram.
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The properties: there are four of them--the first one is that opposite sides are congruent;
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opposite angles are congruent; consecutive angles of a parallelogram are supplementary;
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and the last one is that diagonals bisect each other--those are four properties.
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So, using those properties, let's continue with our examples.
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The first one: complete each statement about the parallelogram.
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I know that I have a parallelogram; BC (there is that side) is congruent to what?
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You are going to complete the statement: BC is congruent to what?--the side opposite, so AD.
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Now, I know that this sounds really simple; it looks like the only other side that could be congruent to BC is AD.
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But sometimes parallelograms are not so stretched out like that; sometimes this side and this side could look a lot closer in length.
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Just don't assume; if it is just a parallelogram, even if side BC and this side right here, CD, look like they are congruent, you can't assume that.
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So, all we can assume from parallelograms when it comes to their sides is that opposite sides are congruent.
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AE is congruent to BE? No, AE is congruent to DE? No, AE is congruent to CE.
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This diagonal is just cut in half; so AE is congruent to CE, the other half of the same diagonal.
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AD is parallel to what? AD is parallel to BC.
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And then, angle BCD is congruent to the angle opposite.
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Now, we are just talking about the whole angle, not just these parts.
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So then, the whole angle, BCD, is congruent to angle D (since B and D are corresponding)...DAB.
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That is Example 1; the next example is to find the values of x, y, and z of the parallelogram.
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Here is x; there is y; and there is z.
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Since I know that opposite sides are parallel and congruent (these sides are congruent;
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these sides are congruent), I can say that z...let's just say x =, y =, and z =; and that way,
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I have all of my answers right here...so I know that z = 11.
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Now, for x, if you look here, remember: we know that opposite sides are parallel.
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So, if I just extend these out, see how these are my parallel lines, and this is my transversal.
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If I were to draw that out again, there is BC; this is AD; and then, here is my transversal, CA; this is C, and this is A.
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Now, if this is parallel, and this is 40, and this is x, what can I say about x?
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If lines are parallel, then alternate interior angles are congruent.
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So, if this is 40, this and this are alternate interior angles, so x has to be 40 degrees.
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And then, for y...well, I know that y and this one are congruent, and angle B, but they don't give me angle B, so I don't know--
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I can't say that y is the same as this angle, because I don't have that angle.
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I have to find y in another way, so let's see...I know that x is 40; this is 40; what is this angle all together?
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This is 65 + 40; this is 105 degrees; I know that consecutive angles are supplementary, so if this is 105, then this has to be the supplement to 105.
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So, the measure of angle y equals 180 - 105; or you can say 105 + the measure of angle y equals 180; you can do whichever.
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Here, this is 75; so the measure of angle y is 75 degrees.
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Again, all I did here to find y is find the whole angle right here, because that is what makes up this angle of this parallelogram;
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and then this angle with this angle is supplementary.
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The next one: Find the distance of each side to verify the parallelogram.
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I need to find the distance of AB and the distance of BC, this distance and this distance,
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and then compare them, because I know that opposite sides are congruent.
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So then, as long as the distance of A to D, this right here, and this, are congruent,
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and then this and this are congruent, then that is verifying that it is a parallelogram.
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Now, they are just telling you that it is a parallelogram; you are just verifying.
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You are just using the distance formula to just show...you are just writing down...
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now, you know that it is going to be the same, because it is a parallelogram.
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So, they are just asking you to verify that.
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The reason why I am saying this is because, in the next lesson, we are actually going to prove parallelograms.
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We are going to use the distance formula and other theorems and such to actually prove that a quadrilateral is a parallelogram.
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But this is not proving; you are not proving that it is a parallelogram--you are just verifying,
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meaning that it is a parallelogram, but you are just showing it; just actually write the numbers to show that it is a parallelogram.
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The distance formula, we know, is the square root of (x₁ - x₂)...and this doesn't mean x squared,
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or anything like that; it is just the first x, minus the second x, squared; plus y₁, the first y, minus the second y, squared.
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So then, let's find the distance of AB; AB is going to be the square root of...
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Oh, before we begin, let's actually write out the points of what these are.
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B is 1, 2, 3; that is (-3,2); C is 1, comma, 1, 2, 3; D is at 1, 2, 3, comma, -1, -2; and A is at -1, -1, 2, 3.
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That way, I have all of my points.
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So, if I am trying to find the distance of A to B, then I am going to have to use these points.
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x₁ (it doesn't matter which one you use) is -1, minus -3, squared, plus y₁, the first y, -3, minus 2, squared.
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-1 minus -3; now, if you remember from algebra, minus a negative makes a plus.
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If you have two negative signs next to each other like that, that just makes both of them into a plus.
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-1 + 3 is 2; that is 2 squared; plus...-3 - 2; that is -5, squared; this is 4 plus 25; and this is the square root of 29.
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And this we can just leave like that, unless your teacher wants you to round it to the nearest hundredth, or some decimal number.
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Then you would have to change this to a decimal.
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Otherwise, you can just leave it like that; I am pretty sure that you can probably just leave it like that.
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Then, I am going to find the side opposite to show that they are congruent: CD.
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CD is, let's see...we can use whichever number, so let's do 3, the first x, minus...we will call that the second x, squared,
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plus the first y, minus the second y, squared; so here, this is 3 - 1, is 2; that is 2 squared, plus -2 - 3 is -5, squared;
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that is 4 + 25, which is the square root of 29.
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Since these are congruent, we know that these sides are congruent.
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Let's do BC: BC = √[(-3 [the first x] - 1)² + (2 - 3)²].
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The square root of...-3 - 1 is -4, squared, plus...this is -1, squared; equals...16 + 1, which equals the square root of 17.
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Then, AD, the side opposite that, is going to be -1 - 3, squared, plus -3 - -2, squared;
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the square root of...-1 - 3 is -4, squared; plus...minus a negative...they both make a plus, so this is -1, squared;
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4 times 4 is 16, plus 1, which is the square root of 17.
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Now, see how this is equal to this, and this is equal to this.
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So then, I know that AB is congruent to CD, and BC is congruent to AD.
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OK, for the next example, we are going to use the same diagram, the same parallelogram.
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And we are just going to verify that it is a parallelogram by using slope.
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Now, how would we use slope to verify the parallelogram?
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If I find the slope of BC and the slope of AD, they are parallel; we know that they are parallel, because it is a parallelogram.
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So, if we find the slope of this, and we find the slope of this, they should be the same,
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because we know that two lines, when they are parallel, have the same slope.
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The same thing works here: we are going to find the slope of this, and find the slope of this, to verify that it is a parallelogram.
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Now, if you want, you can find the points of each one of these, like we did in the previous example.
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Find the coordinates of each; and you can use the slope formula.
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The slope formula is y₂ - y₁, meaning the difference of the y's, over the difference of the x's.
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You could do it that way, using the point of this and using the point of that, and then solve it that way.
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Or, since we actually have a coordinate plane, and we have all of the points there, we can just do rise over run,
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meaning we can just count how many we are rising and how many we are running.
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So then, the slope of BC: to go from this point to this point, you are going to rise one,
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meaning you are going to go up one (and rise just means how many you are going to go up and down).
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Run is how many you are going to go left and right.
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We are going up 1; since we are going up 1, that is a positive 1, over...1, 2, 3, 4; and that is going to the right, so we know that it is a positive number.
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Whenever you go up, it is a positive, because remember: the y-axis...see how when you go up, you are going positive.
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But when you go down, you are going negative; see how these numbers are all negative.
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For the x-axis, the same thing happens: when you go right, you are going towards positive numbers;
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when you go to the left, you are going towards the negative numbers.
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So, if you go up, you are going positive; if you go right, you go positive; going down is negative; left is negative.
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So, we went up; that is a positive 1; and 1, 2, 3, 4--you went to the right 4; that is a positive 4.
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That means that the slope of BC is 1/4.
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For CD, we are going 2 to the right (that is 2), over...how many are we going down?...1, 2, 3, 4, 5.
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That is a negative 5, because we went down 5; the slope of CD is -2/5.
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Now, it doesn't matter if you go from D to C--it will be the same thing.
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If we count from D up to C, we are going to go up 5 (that is a positive 5)...
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Oh, I'm sorry; I did it the wrong way.
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Let's see, let me just try to draw this over again; OK.
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Let's try that again: I am going from C to D.
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Now, I did my rise and run; I did it run over rise--now, be careful not to make that same mistake, OK? I'm sorry about that.
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Now, we are going run...if we go and run 2, I have to write that as my denominator, 2,
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because that is the run; that is the bottom part; that is the denominator.
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And then, I am going down 5; so that is a rise.
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It is actually more helpful if you just do the rise, the up and down, first.
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So then, I would have to go down first, and then to the right; that would be a lot easier, and I wouldn't have made that mistake.
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So then, it is better to just go down...rise first (meaning up and down first), and then you go left and right.
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Here, you can go down 1, 2, 3, 4, 5; that is -5, over 2; that is a positive 2.
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The same thing happens if I go from D to C; I am going to go up 5 (that is a positive 5), over left 2 (that is a negative 2).
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It is the same exact thing.
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Now, the slope of AD will be...let's go from A to D; I am going to go up 1; that is a positive 1, over 1, 2, 3, 4; that is a positive 1/4.
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See how they are the same.
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Then, the slope of AB is going to be...I go from A to B.
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So then, that would be from A..."rise"--going up 1, 2, 3, 4, 5, positive 5, over left 2; that is negative 2; and that is the same as that.
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So then again, I am going to show that this is parallel to this; they are parallel.
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And this is the same as this, which means that these are parallel.
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We are just verifying that they are parallel, and that this is, in fact, a parallelogram.
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In the next lesson, we are actually going to do pretty much the same thing,
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except we are going to prove that they are parallelograms.
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Using the properties, and using the theorems, we are going to actually prove that a quadrilateral is a parallelogram.
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That is it for this lesson; we will see you soon--thank you for watching Educator.com.