WEBVTT mathematics/geometry/pyo
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Welcome back to Educator.com.
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In the next lesson, we are going to continue on with parallelograms.
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And we are going to go over, more specifically, squares and rhombi.
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First, the rhombus: now, *rhombus* and *rhombi* are actually the same thing.
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Rhombus is singular, so when you have only one, then it is a rhombus; when it is plural--you have more than one--then it is actually called rhombi.
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So then, if you hear "rhombus" or "rhombi," then you are talking about the same thing.
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Now, what is a rhombus? A rhombus is a quadrilateral with four congruent sides.
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Here is an example of a rhombus: now, more specifically, a rhombus is a special type of parallelogram.
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You can say that a rhombus is a parallelogram with four congruent sides.
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This right here, this property, is very specific to the rhombus.
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Now, to continue our flowchart, if we have a quadrilateral, quadrilateral goes down to parallelogram;
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and then, parallelogram...we went over the rectangle; that was a special type of parallelogram;
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and then now, we are going to go over the rhombus: it is another special type of parallelogram.
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Now, because the rhombus is a special type of parallelogram, all of the properties of a parallelogram now apply to the rhombus,
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just like when we went over rectangles: since a rectangle is a special type of parallelogram,
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all of the properties of parallelograms apply to the rectangle.
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The same happens with a rhombus; all of the properties apply to the rhombus, as well.
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We went over one property that is very specific to the rhombus; it is all of the properties of the parallelogram, plus four congruent sides.
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Now, there are also a couple of properties that have to do with the diagonals of a rhombus.
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The first one: the diagonals of a rhombus are perpendicular.
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Now, the property on diagonals for a parallelogram is that they just bisect each other.
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They bisect each other; so we know that diagonals bisect each other--that is the very general property of the parallelogram.
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Then, for the rectangle, it became that they bisect each other, and they are congruent.
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And then now, the diagonals of a rhombus bisect each other and are perpendicular.
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Let's say I have a rhombus that...here are my diagonals; we know that diagonals bisect each other.
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That means that this diagonal and this part are congruent.
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And then, the other two halves are congruent to each other.
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And then, this one, the one that is a little more specific to the rhombus, is that they are also perpendicular.
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So, if it is a rhombus, then the diagonals are perpendicular.
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And this is the theorem that goes with that; so if the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.
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This property is very specific to the rhombus, meaning that this property only applies to the rhombus--
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so much so that you can actually use it to prove that it is a parallelogram.
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So, if you can prove that the diagonals are perpendicular, then you can prove that that parallelogram is a rhombus.
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Any time you see anything that is perpendicular--diagonals being perpendicular--you know automatically that that is a rhombus--
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of course, as long as it is a parallelogram (it is a parallelogram with diagonals perpendicular--then, a rhombus).
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The next property that has to do with the diagonals of a rhombus is that they bisect opposite angles.
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There is my rhombus; there are my diagonals; we know that it is perpendicular; we know that the sides are congruent.
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But then, this one now says that the diagonals bisect opposite angles.
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So then, it is bisecting those angles, meaning that this angle is now cut into half.
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And then, these angles are bisected, and these angles are bisected, so that, when they bisect a pair of opposite angles,
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the opposite angles are also congruent, because we know that opposite angles are congruent,
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because that is a property of a parallelogram: Opposite angles are congruent.
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This whole angle is congruent to this whole angle; so this is cut in half, and each of these halves are also congruent to each other.
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Again, there are two more properties of the rhombus, in addition to all of the properties of a parallelogram.
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#1: Diagonals are perpendicular; #2: Diagonals bisect opposite angles.
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Using those properties, let's talk about finding the missing value.
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Again, here is our rhombus; and you know that angle 1 is congruent to angle 2,
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because this diagonal bisects the angles, because it is a rhombus.
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That means that I can just make the measure of angle 1 equal to the measure of angle 2: 2x + 6 = 3x - 19.
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I am going to solve for x by subtracting the 2x over to the other side; I am going to get 6 = x - 19.
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I am going to add 19 to the other side, and I get 25 = x; so there is my x.
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Now, if I need to find the measure of angle 1, I am just going to plug it in; so the measure of angle 1 equals 2(25) + 6.
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So, the measure of angle 1 is...this is 50, plus 6 is 56.
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Now, you know that the measure of angle 1 is 56; the measure of angle 2 is also going to be 56.
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But just to check your answer, you can find the measure of angle 2: 3 times...x is 25...minus 19.
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3 times 25 is 75, minus 19...that is 56.
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So then, we know that our answer is correct.
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Now, next is the square: squares are actually very, very special.
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How are they special? If you look at the definition, a quadrilateral with four right angles and four congruent sides,
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we already heard these properties--we have already talked about these properties before.
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If we break it down, four right angles--what has that property of four right angles?
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We know that that belongs to the rectangle; this is the rectangle's property, four right angles.
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And then, a quadrilateral with four congruent sides--that one belongs to something else, too, and that is the rhombus.
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A rhombus is a quadrilateral with four congruent sides.
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So, that means that a square is made up of the rectangle and the rhombus, which is why a square is a special type of both.
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It has the properties of both the rectangle and the rhombus.
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Continuing with our little flowchart: a quadrilateral goes down to a parallelogram, and then a parallelogram...
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we went over two types, the rectangle and...we just went over...the rhombus;
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and we know that a square is a special type of rhombus and rectangle; and that is the square.
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Remember that a rectangle has all the properties of a parallelogram, plus its own.
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And then, a rhombus has all of the properties of a parallelogram, plus its own properties.
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So then, the square has all of the properties of a rectangle (since it is a type of rectangle);
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it has all of the properties of a rhombus, which means that it also has all of the properties of a parallelogram.
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So, a square is made up of all of these above; a square is just a special type of everything.
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Now, here is that chart again; but let's actually go over each of the properties.
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We know that a parallelogram is two types, rectangle and rhombus; and then, a square is a type of rectangle and rhombus.
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But what about their properties?--let's go over their properties again.
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A parallelogram, we know, has...the definition of a parallelogram says...two pairs (let me write that out) of opposite sides parallel.
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And this is more of the definition of a parallelogram: two pairs of opposite sides parallel.
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The properties: two pairs of opposite sides are congruent; two pairs of opposite angles are congruent;
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and the diagonals bisect each other; and then, one more--consecutive angles are supplementary.
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That is everything that has to do with a parallelogram.
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Now, the rectangle: we know that a rectangle has...
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Instead of listing all of these out, because we know that rectangles have all of the properties of a parallelogram,
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instead of writing all of these out, we will just say "all properties of parallelogram."
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That means that it includes all of this.
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We know that it has four right angles, and then, what about their diagonals? Diagonals are congruent.
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Those are the properties of a rectangle.
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For the rhombus, again, it has all properties of a parallelogram, and then, four congruent sides;
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and then, their diagonals are perpendicular, and the diagonals bisect the angles.
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Those are the properties of a rhombus; now, what about a square?
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I know that a square has all of the properties of a parallelogram, so I am going to write that out: "all of parallelogram;
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all of the properties of a rectangle"; and then, "all properties of the rhombus."
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A square doesn't really have anything that is specific to its own; it just takes on all of the properties of everything else; that is a square.
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That is why it is a little bit special; it is just a mixture of everything.
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That is a summary chart; now, above this, we know that it is a quadrilateral, just a four-sided figure;
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and then, it is more specific to the parallelogram, and so on down.
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Let's now go into our examples: for our first one, we have this table, and all of the properties of diagonals.
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So, you are going to write "yes" or "no" in each of the boxes, depending on if the diagonal property applies to that type of quadrilateral.
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We have our parallelogram, rectangle, rhombus, and square.
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The first property: The diagonals bisect each other.
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Now, try to remember what property that is for--"the diagonals bisect each other."
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That is a property of a parallelogram; so this one would be "yes."
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Now, if that one is "yes," that means that this one applies to all of the other ones, because rectangles have all of the properties
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of a parallelogram; so does a rhombus, and so does a square; this would be "yes," "yes," and "yes."
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Now, the next one: Diagonals are congruent.
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Well, I know that not all parallelograms have congruent diagonals; so then, this would be "no."
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Or you can just leave it blank, just so you can see which ones are actually "yes"; it is just easier to see.
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Let's look at this: what about rectangles--do rectangles have congruent diagonals?--and this one is "yes."
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Does the rhombus have congruent diagonals?--this one is "no"; it does not.
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The diagonals of a rhombus are perpendicular to each other, and they bisect the angles, but they are not congruent to each other.
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That is not a property of a rhombus.
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Now, if you ever get confused, just draw it out; and you know that, if you had to walk from here to here,
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that just seems a lot further than walking from here to here; the distance looks a lot longer this way than it does this way.
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So, they don't seem congruent, and the same thing with a parallelogram; so then, this would be "no."
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But what about a square? Squares have all of the properties of the rectangle.
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So, whatever applies to the rectangle also applies to the square, so this one is "yes."
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Then, the next one: Each diagonal bisects a pair of opposite angles, meaning it does this.
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Now, we have only gone over this one time, because it is very specific to one thing, and that is the rhombus; it is a property of the rhombus.
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Now, again, squares have all of the properties of the rhombus; so then, this will also be "yes."
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And then, the last one: The diagonals are perpendicular--which one is this for?
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This one is also for the rhombus; the rhombus had both of these two diagonal properties, so this one is "yes";
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and again, squares have all of the properties of the rhombus, so this one is also "yes."
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And then, all of the blanks will just be "no"; so you can just fill it in with "no" if you have it written down.
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Otherwise, this way, you can just see what is "yes" and what is "no."
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OK, the next example: we are going to use the rhombus ABCD to find the missing value.
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We have two problems; this is a problem, and this is another problem.
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The first one: The measure of angle ABC is 120; find the measure of angle ACB.
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We are given that this one is 120, and we don't know this one right here; this is what we are looking for, right here--this little part, angle ACB.
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Let's see: we know that this is 120; now, we can do this a couple of ways.
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We can say that, since this is 120, we can use a property of the parallelogram, because a rhombus has all of the properties of a parallelogram.
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We can use the property of a parallelogram that says that consecutive angles are supplementary.
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So then, if angle ABC is 120, then that would mean that angle BCD is the supplement to that; so that would be 60.
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This whole thing is 60; and then, since it is a rhombus, we know that the diagonals are perpendicular, and that it bisects the angle.
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These are cut into two equal parts; so if this whole thing is 60, then this has to be 30--it has to be half.
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The measure of angle ACB is 30 degrees.
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Now, another way to find that is to say that, if this whole thing is 120, then this would be 60;
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this is a right angle; then, this is a triangle, so this would be a 30-60-90 degree triangle,
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because the three angles of a triangle add up to 180.
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A 30-60-90 triangle is a special right triangle; you just know that this is 90, so that means that these two angles have to be 90.
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That is another way to find this angle measure.
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Now, the next one: The measure of angle AED, this one right here, is 4x + 10; find x.
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That is all that they are going to give you--that it is 4x + 10.
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But since we know that this is perpendicular (remember: the diagonals are perpendicular), this angle measure is 90.
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So, make 4x + 10 equal to 90, since that is the angle measure here; and then, we know that it is equal to 90, so 4x = 80; x = 20.
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The next example: Determine if each statement is always, sometimes, or never true.
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If all sides of a quadrilateral are congruent, then it is a square.
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Well, all of the sides being congruent--that is a property of a rhombus.
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And a square is a special type of rhombus, but not all rhombi are squares.
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Sometimes a rhombus is just a rhombus; it doesn't have to always be a square, so this one would be "sometimes."
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If all of the sides of a quadrilateral are congruent, then it is always a rhombus; but a rhombus is sometimes a square.
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If a quadrilateral is a parallelogram, then it is a rhombus.
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Well, a parallelogram is not always a rhombus; it is sometimes a rhombus, because a parallelogram could also be a rectangle.
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So, a parallelogram is only sometimes a rhombus.
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The next one: If a quadrilateral is a rhombus, then it is a parallelogram.
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Well, yes, because a rhombus is always a parallelogram--it is a special type of parallelogram.
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Just like we said, if we have a dog, a type of dog is, let's say, Chihuahua.
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Well, a Chihuahua is always a dog; since a Chihuahua is a type of dog, a Chihuahua is always going to be a dog.
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Just like that, a rhombus is a special type of parallelogram; so a rhombus is always a parallelogram; so this is going to be "always."
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The last one: If a quadrilateral is a square, then it is a rectangle.
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Is a square always a rectangle? It is almost the same as #3: a rhombus is a type of parallelogram; therefore, it is always a parallelogram;
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a square is a special type of rectangle; therefore, a square is always a rectangle.
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Now, if this is still confusing, we can use that flowchart that we did to help us with this.
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If I say, let's say, "parallelogram," instead of writing it all out--we know that that is a parallelogram--
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a parallelogram became a rectangle, and it became a rhombus; then this and this became a square; on top of that is a quadrilateral.
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Now, if it is going from "if" to "then" downwards, then the answer is going to be "sometimes."
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If it is going from "if" to "then" upwards, then it is "always"; and if it is going side-to-side, then it is "never."
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Let's look at these again: #1: If all sides of a quadrilateral are congruent, then it is a square.
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We know that they are talking about the rhombus, basically saying that, if it is a rhombus, then it is a square.
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Well, isn't this sometimes? A rhombus is only sometimes a square; a rhombus can just stay a rhombus, so that is "sometimes,"
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because it is going downwards, from a rhombus to a square.
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The next one: If a quadrilateral is a parallelogram, then it is a rhombus.
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See how it is going downwards from a parallelogram to a rhombus; so if it is going down, it is "sometimes."
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Down is sometimes; up is always; and side-to-side means never.
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If a quadrilateral is a rhombus, then it is a parallelogram.
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It is going upwards; then it is always, because this is a special type of that.
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And then: If a quadrilateral is a square, then it is a rectangle.
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See how it is going upwards? That is always; so you can use this to help you with this.
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The next example: Determine whether quadrilateral ABCD is a parallelogram, a rectangle, a rhombus, or a square with the given vertices.
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With these vertices, we are going to have to do a few things to see what the most specific type of quadrilateral is.
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We know that, first of all, to figure out if it is a parallelogram, we have to find the slope.
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Slope will help us with anything that has to do with being parallel or perpendicular, which has to do with both of these, this one and this one.
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And then, for the rhombus, as long as we can show that it is a rectangle and a rhombus,
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then we can just automatically say that it is a square, too, if it is both a rectangle and a rhombus.
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Now, how do you show if it is a rhombus?
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Well, you know that a rhombus has four congruent sides, so we would just have to use the distance formula to show the length of each side.
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If parallelogram works, then we have to continue with rectangle; if rectangle works, we are going to continue with rhombus.
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If this doesn't work, then it would be a rectangle; but let's say rectangle and rhombus worked--then it would be a square,
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because "square" means that it is both rectangle and rhombus.
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Oh, and then, also, for a parallelogram, if rectangle doesn't work after parallelogram, then we would move on to rhombus.
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We would have to actually show all three of these, but we don't have to show the square, because the square is just all of the above.
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The slope of AB: again, it is the difference of the y's (so I am going to take the first y, which is -6,
00:30:11.900 --> 00:30:26.800
and subtract the second y), and then take the first x, and subtract the second x.
00:30:26.800 --> 00:30:37.600
A minus negative: this becomes a plus; this is -3 over 4.
00:30:37.600 --> 00:30:44.600
Then, if you need help to determine which sides have to be parallel to what, and what sides have to be perpendicular to what,
00:30:44.600 --> 00:30:49.600
just draw any quadrilateral, ABCD; you know it has to be in this order.
00:30:49.600 --> 00:31:00.600
It doesn't matter if you do ABCD or if you do ABCD...it doesn't matter, as long as you know that it is in the order.
00:31:00.600 --> 00:31:27.900
Then, the slope of BC: -3 minus -6, over 5 minus 1: this, again, becomes positive; this is 3/4.
00:31:27.900 --> 00:31:37.300
Well, that is strange; let's see--let's double-check this.
00:31:37.300 --> 00:32:03.500
For AB, -6 - -3 became -3; 1 - -3 became 4; then for this one, y₂ is -3, minus y₁, which is -6;
00:32:03.500 --> 00:32:18.300
and then, x₂, which is 5, minus x₁, which is 4...this is 3/4; so it seems like it is correct.
00:32:18.300 --> 00:32:28.400
Now, here, this slope is -3/4, and this slope is positive 3/4.
00:32:28.400 --> 00:32:37.000
That means that they are not perpendicular to each other, because, in order for this side and this side to be perpendicular,
00:32:37.000 --> 00:32:40.000
their slopes have to be the negative reciprocal of each other.
00:32:40.000 --> 00:32:48.700
Then, if this is going to be -3/4, then this has to be positive 4/3, but it is not.
00:32:48.700 --> 00:32:58.000
So, automatically, I can conclude that this AB and this are not perpendicular.
00:32:58.000 --> 00:33:05.900
If it is not perpendicular, then I can cross out some of these: I can cross out the rectangle, and I can cross out the square,
00:33:05.900 --> 00:33:13.500
because in order for it to be a square, this has to be true; rectangle has to be true, and rhombus has to be true.
00:33:13.500 --> 00:33:19.800
Now, let's continue, because we still have two other options.
00:33:19.800 --> 00:33:33.300
I am going to go for my next one, the slope of CD: 0 - -3 over 1 - 5.
00:33:33.300 --> 00:33:47.300
So again, this becomes 3/-4; so that means that AB is parallel to CD.
00:33:47.300 --> 00:33:53.000
Does that make that a parallelogram?--no, not yet: it has to be two pairs of opposite sides being parallel.
00:33:53.000 --> 00:33:59.700
Or we can say that AB is congruent to DC, if you remember from the section on the parallelogram--
00:33:59.700 --> 00:34:05.400
on the properties, or the theorems to prove a parallelogram.
00:34:05.400 --> 00:34:11.000
There is one theorem; it is not a property of a parallelogram--it is a theorem that says that,
00:34:11.000 --> 00:34:24.700
if you can prove that one pair is both parallel and congruent, then it is a parallelogram; that is another way you can do it.
00:34:24.700 --> 00:34:44.200
Now, the last one, CB and then AD: 0 - -3, over 1 - -3, becomes 3/4.
00:34:44.200 --> 00:34:51.300
So then, BC and AD are parallel, because they have the same slope.
00:34:51.300 --> 00:34:55.900
So automatically, you know that that is a parallelogram; but then, a rhombus is also considered a parallelogram.
00:34:55.900 --> 00:35:13.000
So then, we have to find the distance of each one, the distance of AB (let me do a lowercase d)...
00:35:13.000 --> 00:35:25.200
We know that the distance formula is (x₂ - x₁), the difference of the x's, squared,
00:35:25.200 --> 00:35:33.600
with the difference of the y's, squared, all under a square root.
00:35:33.600 --> 00:35:56.200
For AB: 1 - -3, squared, plus -6 - -3, squared; this is going to be the square root of...
00:35:56.200 --> 00:36:12.900
here is 4 squared; that is 16, plus...this is -3, squared, which is 9; that is going to equal √25, which is 5.
00:36:12.900 --> 00:36:40.100
And then, let's find BC: the distance of BC: this is 5 + 6, squared, plus -3 + 6;
00:36:40.100 --> 00:36:47.000
and I am just making them automatically plus for minus a negative.
00:36:47.000 --> 00:37:06.300
Oh, wait: 5 - 1...I did the y instead of the x...the 5 minus the 1 is going to be 4 squared, which is 16,
00:37:06.300 --> 00:37:13.600
plus 3 squared, which is 9, which is equal to √25, and that is 5.
00:37:13.600 --> 00:37:25.300
And then, we know that this AB is going to be congruent to BC.
00:37:25.300 --> 00:37:42.700
Let's try the other ones: the distance of...what is next?...CD: the square root of 5 - 1, squared...
00:37:42.700 --> 00:37:49.000
Now, notice how I did this one minus this one before I did this one minus this one--it is the same thing,
00:37:49.000 --> 00:37:56.400
as long as, for the next part, for the y's, you do the same order.
00:37:56.400 --> 00:38:05.600
If you are going to do 5 - 1, then for the y's, you have to do -3 - 0, squared.
00:38:05.600 --> 00:38:13.100
That is going to be right here...4² is 16, plus...(-3)² is 9.
00:38:13.100 --> 00:38:17.900
So again, that is the square root of 25, which is 5.
00:38:17.900 --> 00:38:40.900
And then, the distance of the last one, AD: -3 - 1, squared, plus -3 - 0, squared: here we have -4 squared, which is 16,
00:38:40.900 --> 00:38:48.800
plus -3 squared, which is 9; so then, this is the square root of 25, which is 5.
00:38:48.800 --> 00:39:09.700
So, we know that all of the sides are congruent; that means that we have a rhombus, so this is a rhombus.
00:39:09.700 --> 00:39:14.000
That is it for this lesson; thank you for watching Educator.com.