WEBVTT mathematics/geometry/pyo
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Welcome back to Educator.com.
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For the next lesson, we are going to go over polygons.
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We are going to talk about the different types of polygons and the interior and exterior angles of polygons.
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First, let's talk about what is a polygon and what is not a polygon.
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A **polygon** is a closed figure, formed by coplanar segments, such that the sides are non-collinear,
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and each side intersects exactly two other sides at their endpoints.
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Basically, a polygon ("poly" meaning many) is a closed shape (meaning it has to close), each side being a straight lines, and where no sides overlap.
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As long as we have a shape that is closed (nothing open--nothing can get through), with no overlapping, and...
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like this one...see how it is not straight...these are examples of polygons, and these are not.
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Now, it is OK if polygons look funny; if they look like this, that is OK.
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As long as it is a closed figure, each side is a straight line segment, and none of the sides overlap, then it is a polygon.
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So, here, because of that it is not a polygon; because of the overlap, it is not a polygon;
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and because this side right here is not straight, that is not a polygon.
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The two types of polygons are convex and concave; a convex polygon is when all of the sides are on the outside of the shape.
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What that means...maybe if I explain "concave," it will be easier to understand.
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A concave polygon is when two sides go in towards the center of the polygon.
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See how, right here, these two sides are angled towards the center; that would make this concave.
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The same thing happens here: we have this angle going towards the center.
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Think of a cave, like a mountain, or in the rocks; see how it goes inwards, and it creates a little cave?
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So, any time it does that, it is a concave polygon.
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If it doesn't, then it is a convex; so all of the angles are pointing away from the center.
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All of these angles are pointing away from the center, and that is convex.
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And this explanation here: No line containing a side of the polygon contains a point in the interior of the polygon.
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It just means that, if you were to draw each of these sides or extend them into lines, it is not going to cut through the polygon.
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If we make this into a line, it is not going to cut through the inside of the polygon.
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The same thing happens here, and the same thing here.
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With this one, however, if I draw a line, see how it cuts in the polygon; that is what it means--that is what this explanation is saying.
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The easiest way to remember: just think of the cave--it is creating a little space right there, like a cave, so they are concave polygons.
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Now, a **regular polygon** is a polygon with all sides congruent and all angles congruent; it is equilateral, and it is equiangular.
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That shows that it is equilateral, and this shows that it is equiangular.
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Any time that it is equilateral and equiangular, it is a regular polygon.
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And in order for it to be equilateral and equiangular, it has to be convex; you can't have a concave polygon that is equilateral and equiangular.
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Maybe it could be equilateral, but not equiangular.
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The interior angle sum theorem is to figure out the sum of all of the angles inside the polygon.
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If I have a triangle (which is actually going to be the first polygon that we are going to use), I have three angles in the triangle.
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The interior angle sum theorem is going to give me the total, or the sum, or the angle measure, of all three angles combined.
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So, if I have a quadrilateral (four angles), what do all four angles of the polygon add up to?
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First, let's start with triangles; a triangle has three sides...number of triangles: a triangle only has one triangle.
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We are going to talk about this in a little bit, but the number of triangles would just be one.
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The sum of angle measures: we know that all three angles of a triangle add up to 180.
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Next will be a quadrilateral, a four-sided polygon: number of sides: 4.
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Now, if I have a quadrilateral, I have two triangles; so the sum of the angle measures is going to be 360.
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I know that all of the angles added up together in a quadrilateral are going to add up to 360.
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And then, a pentagon is the next one: it has 5 sides; the number of triangles is going to be 3 (let's see if I can draw this: that would be 1, 2, 3).
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The sum of the angle measures: here, every time we add a triangle...every time we have one more side,
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it is like we are adding another triangle in the polygon, and then we add another 180,
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because, for every triangle that exists in the polygon, there is an additional 180.
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We always start with the triangle, because that is the first polygon.
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Then, when you get to quadrilateral, the next one, it is going to be plus 180.
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Then, to get to the next one, we are going to do + 180, which is going to be 540;
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so the angle sum of a pentagon is going to be 540 degrees.
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How about the next one, which is a hexagon?--6 sides: 1, 2, 3, 4...
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Again, we just add 180; it is going to be 720; and so on.
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For each triangle that exists, again, it is going to be 180 degrees.
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But what if I ask you for a 20-sided polygon--what is the interior angle sum of a 20-sided polygon?
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There is a formula to go with this, and that is right here.
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Because a triangle has 3 sides, but only one triangle exists; that is 180.
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For every additional triangle, it is going to be an additional 180; so here, isn't this 2 times 180?--because it is 180 + 180, which is 360.
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Here, from a 5-sided polygon, there are three triangles that exist, so isn't that 3 times 180 (180 + 180 + 180, which is 540)?
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So, it is 180 times the 3; here, there are four triangles, so I have to do 4 times 180.
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So, if I want to find a 20-sided polygon, how many triangles exist?
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Well, look at the pattern: 3:1, 4:2, 5:3, 6:4, 20...it is 2 less, so it is going to be 18.
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Now, again, this is to get 360 here; so we just do 2 times 180, which is going to equal 360; 3 times 180...
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the number of triangles times 180...4 times 180 was 720.
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So here, all I have to do is multiply 18 times 180.
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So first, I have to figure out how many sides I have; this is going to be n.
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And then, subtract the 2 to figure out how many triangles exist in that polygon; and then just multiply it by 180.
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So then, 180 times 18...0...this is 64...8 + 6 is 14...then put the 0 here; 0, 8, 1; 0, 4, 12...it is going to be 3240 degrees.
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All 20 angles of a 20-gon are going to add up to 3240 degrees.
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Looking at the formula, it is going to be the number of sides; subtract 2 (you are going to solve this out first)
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to figure out how many triangles you have in that polygon; and then just multiply it by 180; that is it.
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It is just the number of sides, minus 2: take that number and multiply it by 180, and that is going to give you the interior angle sum.
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Now, the exterior angle sum theorem ("exterior" meaning outside): whatever you have...it can only be one exterior angle
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from each side or vertex...then if this right here is 1, this is 2, this is 3, this is 4, and this is 5
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(there have to be 5 of them, because there are only 5 sides here; it is a pentagon)--all 5 angles here are going to add up to 360.
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And that is the exterior angle sum theorem; the interior angle sum theorem is different,
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because depending on the number of angles, depending on what the polygon is, the interior angle sum is going to be different.
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The more angles the polygon has, the greater the sum is going to be.
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But the exterior angle sum is always going to be 360--always, always, no matter what type of polygon you have,
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whether you have a triangle (if you have a triangle, it doesn't matter if you measure the exterior angle this way,
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as long as you do the same for each vertex: let's say 1, 2, 3 here; the measure of angle 1, plus...they are all going to add up to 360)...
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Here, the measure of angle 1 + 2 + 3 + 4 + 5 are all going to add up to 360; and that is the exterior angle sum theorem.
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The first example: Draw two figures that are polygons and two that are not.
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You can just draw any type of figures that you want.
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The first two that are polygons...you can just draw...it doesn't matter...any type of polygon.
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You can draw something like that, as long as it is closed, each side is a straight line segment, and no sides are overlapping.
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Then, two examples, two figures, that are not polygons would be exactly those things.
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Maybe like that...that would not be a polygon; if I have two sides crossing like that, that is a non-polygon.
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If I have, I don't know, something like this, that wouldn't be a polygon...use any examples that are something like this.
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These are polygons; these are not polygons.
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Moving on to the next example: Find the sum of the measures of the interior angles of each convex polygon.
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The first one is a heptagon: now, a heptagon is a 7-sided polygon; this has 7 sides.
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Remember: if the number of sides is 7 (n is 7), we have to figure out how many triangles.
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Remember: you subtract 2, so the number of triangles is going to be 5; and then, you are going to multiply that by 180.
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Now, the formula itself is going to be that the sum is equal to 180 times (n - 2).
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This is 7 - 2; that is 5; so it is the same thing as 180 times 5.
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Then, you can just do it on your calculator; I have a calculator here; 180 times 5 is going to be 900, so the sum is 900 for a heptagon.
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The next one is a 28-gon; now, once you pass 12-sided polygons, there is no name for it, so you would just write 28 with "-gon."
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This is a polygon with 28 sides, so n is 28; the number of triangles is 26; you are going to multiply that by 180.
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To use the formula, you are going to do 180 times number of sides; that is 28 - 2, so 180 times 26...use your calculator...is going to be 4680 degrees.
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So, all the interior angles of a 28-gon are going to add up to 4680 degrees.
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And the next one, x-gon: now, for this one, we don't know how many sides there are in this polygon.
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So, you are just going to use the formula; and so, we know that n is going to be x; the number of sides is x.
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In the formula, you are just going to replace the n with the x.
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It is supposed to be the number of sides, minus 2; instead, we are going to say x - 2.
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And that would be it; you are just replacing the n with whatever they give you as the number of sides, and that would be x.
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Given the measure of an exterior angle, find the number of sides of the polygon.
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Before we start with these numbers, with these examples here, I want to first use a triangle.
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Now, we know, from the exterior angle theorem, that the sum of the exterior angles is always 360.
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The sum of the measures of all of the exterior angles is going to be 360.
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If I have a triangle, here, here, and here: those are my three exterior angles: 1, 2, 3.
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How would I be able to find the number of sides?
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Well, in this case, how would I find each of these angle measures?
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Wouldn't I have to do 360 degrees, divided by the number of exterior angles?
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This is going to be what?--each of these angles has to be 120.
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Now, again, this is going to be for a regular polygon; for a regular polygon, this is going to be 120;
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if all of these exterior angles have the same measure, then it will be 120 each.
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So, that way, it will total to 360.
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Well, it is like they are giving you the measure of each exterior angle; so how would we figure it out...
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if I said that each exterior angle has an angle measure of 120--each exterior angle of a polygon is 120--find the number of sides.
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Well, you would have to do the same thing: 360 (because that is the total) divided by 120, and that is going to give you 3.
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So, you know that there are three sides here.
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The same thing works for this: 36 is the angle measure of each exterior angle.
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So, if 360 degrees is the sum of all of the exterior angles, divide it by 36 to find the number of sides; you get 10.
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That means that the polygon has 10 sides.
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The same thing works here: 360, divided by 15 degrees (you can use your calculator), is going to give you 24 sides.
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If there are 24 sides, each angle of the sides (because if there are 24 sides, that means that there are 24 angles)--
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each exterior angle--has a measure of 15 degrees, which will then
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(since there are 24 of them) add up to 360 when you multiply these two together.
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The same thing works for this one, x: each exterior angle measure is going to be x degrees--we just have to divide it.
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And since we can't solve that out, this will just be the answer; you are just simplifying it out as much as possible, and that will be it.
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There is nothing else that you can do with that.
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The fourth example: Find the sum of the interior angles of each polygon.
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Now, notice how both of these polygons are not regular polygons; it doesn't look like it is equilateral; it doesn't look like it is equiangular.
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But it is OK, because we are just looking for the sum of all of the interior angles.
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Since it is not regular, we would not be able to find the measure of each angle.
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But instead, we can find the sum--what all of them add up to--because it depends on the polygon, not the type of angles inside the polygon.
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Here, we have 1, 2, 3, 4, 5, 6; so we have a hexagon, a 6-sided polygon.
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And that means, in a 6-sided polygon, that we have 4 triangles.
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Remember: we subtract 2; we get 4 triangles; and then we have to multiply this by 180.
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4 times 180...this is 32...720; that means that the sum of all of the angles inside here is going to be 720.
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Now, if, let's say (I am just going to add to this problem here), this was a regular polygon--
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say that all of the sides are the same, and all of the angles are the same, so it is equilateral, and it is equiangular;
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and I want to find what the measure of each angle is, then.
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Since each of these angles are the same, and I know that all 6 angles together
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are going to add up to 720, how can I find the measure of just one of them?
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Since they are all the same, how can I find the measure of just this one, the measure of angle A, or the measure of 1?
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Since they all have the 720, and they are all the same--they all have the same measure, and there are 6 of them,
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I can just take 720 and divide it by 6; 720/6 is going to give me the measure of each of these angles.
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So then, here, you do 720 divided by 6; each of these angles is going to be 120; 120 here, 120 here, here, here, here, and here.
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And that is only if you have a regular polygon, meaning that all of the angles are the same.
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All of the angles have to be the same for you to be able to divide your angle sum to figure out each of these angle measures.
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The next one: here, this is to find the sum of all of the angles side.
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This is a quadrilateral; we only have four angles; so this is just going to be 180 times 2: let's just use the formula...
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180...n - 2 is the sum; 180...we have four sides, minus the 2, so that means we have two triangles;
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180 times 2, we know, is 360 (I said 360, and I wrote 320).
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Now, again, if all of these angles were the same, were congruent, this is equilateral and equiangular, so it is a regular polygon.
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Then, you would take 360; you can divide it by 4; and that would just be 90 degrees; that is if each of these angles were the same.
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Then, each of them would have a measure of 90; and we know that that would just make this a square, if it was an equilateral, equiangular quadrilateral.
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That would make it a square; then you would know that each of these angles would have to be a right angle.
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But for the sake of just knowing what to do if you have a polygon that is regular--
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not just a quadrilateral, but any other type of regular polygon--you would just take the sum,
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and divide it by the number of angles you have.
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And that is it for this lesson; thank you for watching Educator.com.