WEBVTT mathematics/geometry/pyo
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Welcome back to Educator.com.
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For this lesson, we are going to go over two more types of quadrilaterals, which are trapezoids and kites.
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First, a **trapezoid** is a quadrilateral with exactly one pair of parallel sides.
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We know that this is not a special type of parallelogram, because parallelograms have two pairs of parallel sides.
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A quadrilateral with one pair--that is the key word; if it only has one pair, it is a trapezoid; if it has two pairs, it is a parallelogram.
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A trapezoid, here...now, that is the only requirement for a quadrilateral to be considered a trapezoid.
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It is just that all it has to have is one pair of parallel sides; that is it.
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Now, this right here...each side of a trapezoid has a special name: here, this is called the base;
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this is also called the base; base doesn't mean the side on the bottom--the bases are the two sides that are parallel,
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because we can have a trapezoid that looks like this.
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That would be considered a trapezoid, because these two are parallel.
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It does not mean that this is the base; this is not the base; this is the base, and this is the base.
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It is just that the parallel sides would make it the base.
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And then, the two other sides are called legs: base, base, leg, leg.
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And then, these right here would be considered the base angles--these two--those are all base angles.
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Now, an isosceles trapezoid is a special type of trapezoid.
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An isosceles trapezoid is when the trapezoid has congruent legs (I am going to erase this, because it is just kind of in the way).
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We are actually going to go over the angles next.
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So for now, an isosceles trapezoid is just a trapezoid with congruent legs; when these are congruent, then it is an isosceles trapezoid.
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If they are not congruent, it is just a trapezoid; they are still trapezoids, but again, if the legs are congruent, then it is an isosceles trapezoid.
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Here is the isosceles trapezoid; now, we know what an isosceles trapezoid is--a trapezoid,
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meaning a quadrilateral with one pair of parallel sides, with congruent legs (and these are the other legs).
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These are the two legs that have to be congruent.
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Now, for an isosceles trapezoid, both pairs of base angles are congruent.
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Now, remember how this is the base, and this is the base; so this and this are called the base angles.
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Now, this would be a pair; this one and this one would be considered base angles; that is one pair.
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Then, these two angles are considered the second pair of base angles that are congruent.
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Only for an isosceles trapezoid are base angles congruent.
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The next one: the diagonals of an isosceles trapezoid are congruent; so all of this is congruent to all of that; diagonals are congruent.
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The next part of the trapezoid is the median: the **median** is this middle segment here.
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But it is not just any middle segment; this point right here has to be the midpoint of this side, of the leg;
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and this point has to be the midpoint of this leg, and that would make it the median.
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It is a segment whose endpoints are the midpoints of the legs: leg, leg, midpoint, midpoint,
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and then you draw the segment that connects those midpoints together, and that would be called the median.
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Again, to go over isosceles trapezoids, just to review isosceles trapezoids: we know that the legs are congruent;
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we know that two pairs of base angles are congruent--not opposite pairs of base angles;
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the top two or the two angles that have to do with one base, and then the other two that have to do with the other base.
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Those two pairs are congruent, and then the diagonals are congruent (of an isosceles trapezoid).
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The median is the midpoint of one leg, connected to the midpoint of another leg.
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And the median can be for any trapezoid; it doesn't just have to be for an isosceles trapezoid.
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You can have a trapezoid that looks like that; remember: as long as those are parallel, then it is a trapezoid.
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As long as I have a middle point of that side and the midpoint of this side, and I connect these two, this would be the median.
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How do you find the median? Let's just sketch this out: the midpoint of this side...the median...and then these are parallel.
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The median of a trapezoid is parallel to the bases; this median is going to be parallel to the bases.
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These are the bases, because those two are the parallel sides.
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The median is parallel, and its measure (this is the median, EF) is one-half the sum of the measures of the bases,
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meaning that all of this is base 1 plus base 2, divided by 2.
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So, you are just adding up this base, adding up the other base, and then just dividing that number by 2.
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Now, that might sound familiar to you; if you add up the two, and you divide by 2, that is also the average.
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That is how you find the average; so if you are finding the average of the two bases, that would be the measure of the median.
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So, if you forget this (1/2 times the sum of base 1 and base 2), then you can just remember that as the average of the two bases,
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because average is pretty easy; you know that you have to add them all up and then divide by however many you have.
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In this case, you have two bases; so you add up the two bases and divide it by 2, and that is just the median.
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Now, the kite: most classrooms don't really go over the kite.
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Most classes, most teachers, skip over kites, because we don't really care too much about kites.
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But I am going to go over it, just briefly, and just explain to you what a kite is.
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It is a quadrilateral (we know, because it has four sides) with two pairs of adjacent congruent sides.
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It is not like a parallelogram, where it has two pairs of opposite sides being congruent.
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There are still two pairs of congruent sides, but then, they are adjacent.
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That means that this side and this side are congruent, and then these other remaining two sides are congruent; that is a kite.
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Also, the diagonals are perpendicular; and then, the long diagonal, which is this right here, bisects the angles.
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It is not both diagonals; it is not like the rhombus, where the diagonals bisect all of the angles.
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It is just this right here--it is just these angles that are bisected.
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We know that the two pairs of adjacent sides are congruent; the diagonals are perpendicular;
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and then, only those are congruent, and then those are congruent.
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They are not congruent to each other; just this angle is bisected, and then this angle is bisected.
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And that is pretty much it for kites: again, one more time, a kite is a quadrilateral with two pairs of adjacent congruent sides.
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So, these two are congruent, and then the other two are congruent to each other--not the opposites.
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Diagonals are perpendicular, and then, only this angle and this angle are bisected by the diagonal.
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The longer diagonal bisects each of the angles.
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And that is the kite.
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Let's fill in these lines, going over the different types of quadrilaterals that we went over.
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This top part, right here, is going to be a quadrilateral; so then, this whole thing is on quadrilaterals.
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Then, we went over three different types of quadrilaterals.
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The first one, the one with all of this stuff below it, is parallelograms.
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How do I know that this one is parallelograms?--because I have a lot of things that I can write under here--the different types of parallelograms.
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What are they? The two different types of parallelograms that we went over are rectangle and rhombus.
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Then, what about this right here? This is the square.
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What is this right here? This is another type of quadrilateral that is not a parallelogram, so it doesn't go below the parallelogram; it is the trapezoid.
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And then, we went over a special type of trapezoid: if the legs are congruent, then it would be an isosceles trapezoid.
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And then, a third type of quadrilateral that we went over briefly is the kite.
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And there are no special types of kite, so that is just kite; that is it.
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This will help you for those questions with "always/sometimes/never."
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We know that a quadrilateral is sometimes anything below it; when it goes down, it is "sometimes."
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A quadrilateral is [always/sometimes/never] an isosceles trapezoid--well, it is "sometimes,"
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because sometimes it is a trapezoid, and sometimes that is an isosceles trapezoid.
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A parallelogram is sometimes a square.
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A rhombus is always a parallelogram; a square is always a quadrilateral.
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Remember: if you are going upwards on this chart, then it is "always."
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A kite is always a quadrilateral; a kite always has four sides.
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Now, when you go side-to-side, that is when it is "never," because a parallelogram is never going to be a trapezoid.
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It is either going to have one pair of parallel sides or two pairs of parallel sides, to make it one or the other.
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So, when it goes side-by-side, then it is "never"; a parallelogram is never going to be a trapezoid.
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Or a rectangle is never going to be an isosceles trapezoid, or a square is never going to be a kite.
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Anything that is side-by-side, that is not going up the arrows, is going to be "never"; this will help you with that.
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Let's go over examples: State whether each statement is true or false, based on isosceles trapezoid ABCD.
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Let me just label this out: here is A, B, C, and D; I will label this as E.
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Isosceles trapezoid: that means that we know that these are parallel, and these are congruent.
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We also know that this diagonal is congruent to this diagonal, and we know
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that this pair of base angles is congruent, and then the top pair of base angles is congruent.
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The first one: AC is congruent to BD--now, remember: that is only true for isosceles trapezoids.
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If it is a regular trapezoid, then none of these properties apply to them; it is just isosceles trapezoids.
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Irregular trapezoids are not isosceles trapezoids; the only thing they have is the pair of parallel sides--that is it.
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All of these extra properties are all only for isosceles trapezoids.
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So, AC is congruent to BD, because it is an isosceles trapezoid; this one is true...and this needs that, and so does this.
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The next one: AC is perpendicular to BD.
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Now, there was no property or theorem that went over that.
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It could be, but there is nothing that we went over that says that AC is perpendicular to BD.
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The only thing that we went over off our diagram is that they are congruent.
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So, in this case, this is false; it could be true, but we don't want to say that it is true when it could be false, so we are just going to say that it is false.
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The next one: Angle ABC and angle BCD are supplementary.
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Now, think about it: here is AB, extended; here is BC, extended; remember: these are parallel.
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Then, let's say that BC is a transversal; this is angle ABC, so this angle...and then angle BCD, this angle--are they supplementary?
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Yes, they are, because they are consecutive interior angles (or you can say that they are same-side interior angles).
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When it comes to consecutive interior angles, we know that they are supplementary, as long as these lines are parallel (which we know they are).
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These two angles are supplementary; they are going to add up to 180; this one is true.
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The next example: EF is the median of trapezoid ABCD.
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If it is a median, we know that these are congruent, and these are congruent.
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Now, the reason why I am not marking these parts congruent to these parts is because I don't know that it is an isosceles trapezoid.
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This leg can be different than this leg; so then, I just have to mark this as the median of this whole leg, and then this as the median of this leg, separately.
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So, I know that AD is 5; this is 5; BC is 13.
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Remember: to find the median EF, EF is (let's just write it here) one-half (AD + BC)--
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in other words, AD + BC, divided by 2, or the average of the bases, which are these two.
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AD is 5 (let me write...); EF equals 1/2(5 + 13); so then, this right here is 18, divided by 2 is 9; so EF = 9.
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The next one: BC is 15; EF is 11; find AD.
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Now, they give you the median, and they give you BC, one of the bases; but they are asking for the second base.
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I am just going to use the same formula right here, and I am going to plug in 11 for EF; that equals 1/2...
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what do they give me?...BC, so then...AD...I am going to write it as my variable, plus 15.
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With this, I can either distribute the 1/2, or I can say 11 = (AD + 15)/2.
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And that way, it will be easier, because I can just multiply the 2 over to both sides, and I get 22 = AD + 15.
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Subtract the 15, and I get 7; so then, AD is equal to 7.
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So then, if they give you a median, and they are asking for one of the bases, then you just leave that as your variable.
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The next one: AD is x + 4; EF is 12; and BC--this is BC--is 2x + 2; find x...it is the same thing here.
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EF is 12; that equals...and I am just going to put it over 2, so then AD, which is x + 4, plus BC, 2x + 2, all over 2.
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So then, this becomes 3x + 6, divided by 2; again, multiply this by 2; 24 = 3x + 6; subtract the 6; I get 18 = 3x, and x = 9.
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And then, if you want to double-check your answer, you can just plug it back in.
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So, 9 + 4 is 13...and then, this is 18; this is 20; 13 plus...
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Oh, let's look at this again: we have x + 4 + 2x + 2; 3x + 6, divided by 2...
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we multiply the 2 over, so we have 24 = 3x...OK, I divided it wrong, so it is a good thing that I double-checked.
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This is...x is 6; that means that these would be different.
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It is always good to double-check your answers: 6; this is 10; and this is 12, plus 2 is 14.
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So then, that adds up to 24, divided by 2 is 12; so then, this is the correct answer.
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OK, the next one: Example #3: Complete each statement with "always," "sometimes," or "never."
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Now, this had to do with that flowchart that we did, where, if you are going down on the chart, then it is going to be "sometimes";
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if you are going up on the chart, it is going to be "always"; and if you go side-by-side on the chart, then it is going to be "never."
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So, if you need to write it out, then go ahead; if not, then let's just try to do it without it.
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If you want to draw it out, then that is up to you.
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A square is [always/sometimes/never] a rectangle--again, a square is a type of rectangle, just like a Chihuahua is a type of dog.
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So, a Chihuahua is always a dog, so a square is always a rectangle.
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A quadrilateral is [always/sometimes/never] a rhombus.
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Well, a quadrilateral could be a parallelogram; it could be a rectangle; it could be a trapezoid.
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A quadrilateral can be a lot of different things, so it is only sometimes a rhombus.
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A trapezoid is [always/sometimes/never] a parallelogram.
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This one, we know, is "never," because in order for it to be considered a parallelogram,
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it has to have two pairs of parallel sides; trapezoids only have one pair.
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A parallelogram is [always/sometimes/never] a square.
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Well, a parallelogram could be a rectangle, or it could be a rhombus; a rectangle is not always a square--
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a rectangle can just remain a rectangle; a rhombus can remain a rhombus.
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So, a parallelogram is not always going to be a square--it is "sometimes."
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And it is also going down on the chart, so it is "sometimes."
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A rhombus is [always/sometimes/never] a trapezoid.
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A rhombus is a type of parallelogram; we know that parallelograms and trapezoids are two different things, so it is "never."
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And the last one: Determine if the figure is a trapezoid.
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Now, first, we have to think, "OK, what makes a trapezoid?"
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The only property of a trapezoid is the one pair of parallel sides--only one--that is it.
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All of the other properties, with the legs being congruent, diagonals being congruent, two pairs of base angles being congruent--
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those properties have to do with isosceles trapezoids.
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This one just says "trapezoid"; we are only talking about a trapezoid.
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The only property that would make it a trapezoid is that one pair of parallel sides--that is all we have to do--we don't have to do anything else.
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So, here, we know that, if we were to have one pair of parallel sides--if it is a trapezoid--
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then it has to be these two sides here, because we know that this and this are not going to be parallel.
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So, the slope of this is going to be, remember, rise over run; we are going to count...
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Rise is going up and down; run is going side by side.
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If you go up, remember, it is a positive number; if you go down, it is a negative number.
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If you go to the right, it is a positive number; and if you move to the left, it is a negative number.
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Here, let's go up; we are going to find the slope of this right here.
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And again, we are finding the slope of this one and this one, because, if they are parallel, then they will have the same slope.
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We just have to find the slopes; and if they are the same, then those two will be parallel.
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So then, the slope of this is going to be 1, 2; so that is +2, over 1, 2, 3, 4.
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It is going to the right, so that is positive, so it is positive 4; that means that the slope of this is 1/2.
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Now, if you are still a little unfamiliar with slope, I can also go down, just to show you.
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If I go down 2, that is -2, over...and then I am going to go to the left 1, 2, 3, 4; that is -4, because I went to the left.
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This also becomes 1/2, so it is the same; I just wanted to show that, if you go from this point to this point,
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you are going to get the same slope as when you go from this point to that point.
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Let's look for the slope of this one right here: from here to here, I get...actually, let's go from here to here.
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I am going to go down 3: 1, 2, and then one more--3: that is -3.
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And then, I have to go to the left 1, 2, 3, 4, 5, 6; that is -6, which is 1/2.
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So, this slope is 1/2, and this slope is 1/2; therefore, this is a trapezoid--"yes"--because these two are now parallel.
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So yes, it is a trapezoid.
00:30:42.800 --> 00:30:48.000
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