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# In geometry, the angle of a polygon is formed by two adjacent sides of the polygon. Angles are usually assumed to be in a Euclidean plane with the circle taken for standard with a positive origin, or with the x-axis taken as positive. The measure of an exterior angle of a simple (non-self-intersecting) polygon is less than the sum of the measures of its interior angles. A convex polygon has all interior angles less than 180 degrees. An n-sided convex polygon has (n-2) interior angles less than 180 degrees.

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##### Top Educators ##### Catherine R.

Missouri State University  ##### Kristen K.

University of Michigan - Ann Arbor

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Hey, guys, I'm here to talk about Central and inscribed angles again, but this time, how to find their measure. And how do you define their measure? So, given a circle. Okay, let's call this circle C. Remember, this is an example of a central Angleton angle that has its center at the center of the circle. So let's say this angle right here is, let's say, 100 degrees, then the measure of this ark is also going to have a measure of 100 degrees. Essentially, it's an arc that is created by an angle of 100 degrees. So if I call this a Oh, let's put another point here p than the measure of arc. A P O would equal 100 degrees and vice versa. So if we had a circle and I knew that this particular arc from here, Thio here had a measure of, let's say, 130 degrees than I would know. That angle back is also equal to 130 degrees. So the moral of this story is a central angles measure is equal to the Ark measure, which is different than the length of the ark. More on that later, but also vice versa. Whatever the measure of the Ark is, the central angle is as well. Okay, what if we have now? E was a random triangle there. Let's get rid of that. What if we have now a circle? We've been inscribed angle. And let's say that this angle here is 30 degrees. Notice that it creates an arc on the circle from here to here. This arc is actually going to have that's called Arc A B. This particular arc, a B okay from a to B is actually going to be double the inscribed angle. So if you haven't inscribed angle the ark, it's sub tens is going to be double it. Or, in other words, if I have an angle or part of a circle, I haven't inscribed angle. I know that the ark it sub tens, let's say, is 100 degrees that I know that that inscribed angle is going to be 50 degrees. So that's the relationship between inscribed angles and their associative arcs. So if we want to do some practice and we go to this worksheet here and the goal here is to pardon me identified the measures the missing measures to notice here is we're looking for this angle here. It's a central angle. It opens up to a arc that has a measure of 153. So I guess what this angle measures it also equals 1 53. You, too. Just like this angle here. That's not in question. But this angle here would be 148 degrees. If we wanted to figure out what this arc is. We could do 3 60 minus 1 53 minus 1 48. And so this angle here would be 59 degrees because that's what's left of 360 degrees. That would mean, by the way, that this ark also has a measure of 59 degrees. Okay, so likewise, we know that this arc here has a measure of 116 degrees. This ark here has a measure of 1 29. And then the other arc would be what's left from 3. 60. Okay, moving along to some inscribed angles. If you notice we have this inscribed angle here, Z x. Okay. Notice that that opens up to an ark of Z. Why X that arc, the measure of Z Y X. Okay, is going thio equal. Double 102 It's gonna be 240 degrees because, remember, the inscribed angle of a circle is gonna be half of the market sub tens. So the ark is gonna be double that, which is 204 Yeah, okay, so that that's how that works. Okay, what else do we know here? We also know that if this is 204 then we know that the other arc Z a X is going to be what's left from 363 60 minus 204 is going to get us a measure. So the measure of Arc Z a X is going to be 100 and 56. Okay, now what? That also means is that since that arc is created by this angle, we know that that angle Z Y X is half of 1 56. So if we divide 1 56 divided by two, we get an angle of 78 degrees. So notice to get to what we wanted, we had to actually calculate a few other things and kind of work and directly to get to it. So a za recap. We were given this angle so we knew that this ark was double that. That's where we have the two of four from 3. 60 minus that got us the rest of the ark. We're looking for 1 56 and then it's inscribed angle would be half of that over here. We have a very special situation. Take that back. Almost a special situation. If this were 90 degrees, we have a very, very special situation. But we'll get to that in a second. What we know now, though, is this questionable. This this'll arc in question is gonna be double the inscribed angle. So two times 86 we're going to get a measure of 172 degrees. Comment I was going to make was, If you have a circle and you have a diameter, then if you think about it, If this angle here is 1 80 then the inscribed angle is gonna be half of that, which is 90. So any triangle that's inscribed within a semicircle automatically be a right triangle because it will sub tend half of the circle and therefore the inscribed angle's gonna be half so you could put the point here, draw that triangle and you still have a right angle. You could put the point right here. Draw that triangle, and you still have a right angle. So as long as that triangle has its high pot noose or a side that is the diameter, it's gonna force that triangle to be a right triangle. Okay, let's take a look at this one. This is a great question. If we're looking for, um, what the value of this angle is, notice that it's sub tens or opens up to this arc length of 62 degrees. That would mean that the question mark is gonna be half of that or, in other words, 31 degrees. But I want to add on to this. There's something other, something very special that we also know. We also this angle here is going to be 31 degrees as well, because it opens up to the same mark so automatically, those two angles, we're gonna be congruent because they open up to the same mark. Yeah, so this is a little bit a couple examples of how you would measure or find the measure of essential angle or an ark based upon its central angle, or how to find the measure of an inscribed angle or its are based upon one another. So if you're a central angle, your equal to the measure of your ark and vice versa. If you're a inscribed angle, you're half the measure of the R Q sub tent, and therefore the Ark is going to be double the angle it is created by. Webster University
##### Top Geometry Educators ##### Catherine R.

Missouri State University  ##### Kristen K.

University of Michigan - Ann Arbor