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Current time:0:00Total duration:14:16

Inscribed angle theorem proof

CCSS.Math:

Video transcript

what I want to do in this video is to prove one of the more useful results in geometry and that's that an inscribed angle an inscribed angle is just an angle whose vertex sits on the circumference of the circle so that is our inscribed angle I'll denote it by Sai I'll use Sai for inscribed angle and angles in this video that Sai the inscribed angle is going to be exactly one half of the central angle that subtends the same arc so I just used a lot of fancy words but I think you'll get what I'm saying so this is Sai it is an inscribed angle it's it's its vertex sits on us or on its on the circumference and if you draw out the two rays that come out from this angle or the two chords that define this angle it intersects the circle at the other end and if you look at the part of the circumference of the circle that's inside of it that is the arc that is subtended by Sai so it's all very fancy words but I think the idea is pretty straightforward this right here this right here is the arc arc let me write it this way arc subtended subtended by Sai or sais that inscribed angle right over there the vertex sitting on the circumference now a central angle is an angle is an angle well the vertex is sitting at the center of the circle so let's say that this right here I'll try to eyeball it that right there is the centre of the circle and so let me draw a central angle that subtends this same arc so that looks like a central angle subtending that same arc just like that let's call this theta so this angle is Sai this angle right here is Theta and what I'm going to prove in this video is that Sai is always going to be equal to one half of betta so if I were to tell you that psy is equal to I don't know 25 degrees then you would immediately know that theta must be equal to 50 degrees or if I told you that theta was 80 degrees then you would immediately know that psy was 40 degrees so let's actually prove this so let me clear this so a good place to start or the place I'm going to start is a special case I'm going to draw a inscribed angle but one of the chords that define it is going to be the diameter of the circle so this isn't going to be the general case there's going to be a special case so let me see this is the center right here of my circle trying to eyeball it and we drink it better that looks Center looks like that so let me draw a diameter so the diameter looks like that and then let me define my inscribed angle this diameter is one side of it and then the other side maybe is just like that so let me call this right here sigh if that sigh this length right here is a radius right that's our radius of our circle and then this length right here is also going to be the radius of our circle going from the center to the circumference the circumference is defined by all of the points that are exactly a radius away from the center so that's also a radius now this triangle right here is an isosceles triangle it has two sides that are equal two sides that are definitely equal and we know that when we have two sides being equal their base angles are also equal so this will also be equal to sy you might not recognize it because it's tilted up like that but I think many of us when we see a triangle that looks like this a triangle looks like this if I told you this is our and that is our that these two sides are equal and if this is sy this a little bit neater this is side then you would also know that this angle is also going to be side base angles are equivalent on an isosceles triangle so this is sigh that is also sigh now let me look at the central angle this is the central angle subtending the same arcus let's highlight the arc that they're both subtending this right here is the arc that they're both going to subtend so this is my central angle right there theta now if this angle is Theta what's this angle going to be this angle right here well this angle is supplementary to theta so it's 180 minus theta right when you add these two angles together you go 180 degrees around or you kind of form a line they're supplementary to each other now we also know that these three angles are sitting inside of the same triangle so they must add up to 180 degrees so we get sy sy this side Plus that sy plus sy Plus this angle which is 180 minus theta plus 180 minus theta these three angles must add up to 180 degrees there are three angles of a triangle now we could subtract 180 from both sides subtract 180 from both sides sy plus I is - sigh - theta is equal to 0 add theta to both sides you get 2 sy is equal to theta multiply both sides by 1/2 or divide both sides by 2 you get sy is equal to 1/2 of theta so we just proved what we set out to prove for the special case for the special case where our inscribed angle is defined where one of the the Rays if you want to view these lines as rays we're one of the Rays that defines this inscribed angle is along the diameter it forms or the diameter forms part of that ray so this is a special case where one edge is sitting on the diameter so this could apply so already we could this we could generalize this so now that we know that you know if the is 50 that this is going to be a hundred degrees and likewise right whatever psy is or whatever theta is size is going to be one half of that or whatever psy is theta is going to be two times that and now this will apply for anytime we can use this we can use this notion anytime let me clear this anytime that so let me so just using that result we just got we can now generalize it a little bit although this won't apply to all inscribed angles let's have an inscribed angle that looks like this let's have an inscribed angle that looks like this so in this situation the center you can kind of view it as it's inside of the angle right that's my inscribed angle and I want to find a relationship between this inscribed angle and the central angle that's subtending the same arc so that's my central angle subtending the same arc well you might say hey gene none of these ends or these these chords that define this angle neither of these are diameters but what we can do is we can draw a diameter if this is the center is within these two chords we can draw a diameter we can draw a diameter just like that if we draw a diameter just like that if we define this angle as psy one that angle as sy2 clearly size is the sum of those two angles and we call this angle theta one and this angle theta two we immediately know that just using the result I just got since we have one side of our angles in both cases being a diameter now we know that we know that psy one is going to be equal to 1/2 theta one and we know that sy 2 is going to be 1/2 theta 2 sy 2 is going to be 1/2 theta 2 and so sy which is sy 1 plus I 2 so sy 1 plus let me write the side a little better sy 1 plus sy 2 is going to be equal to these two things half theta one plus one half theta 2 psy 1 plus I 2 this is equal to that the first inscribed angle that we want to deal with just regular sigh that's psy and this right here this is equal to 1/2 times theta 1 plus theta 2 what's theta 1 plus theta 2 well that's just our original theta that we were dealing with our original theta so now we see that psy is equal to 1/2 theta so now we've proved it for a slightly more general case where our Center is inside is inside of the two rays that define that angle now we have to we still haven't addressed a slightly harder situation or a more general situation where me where if this is the center of our circle and I have an inscribed angle whose where the center isn't sitting inside of the 2 chord so let me draw that so let's say I have that's going to be my vertex and I'll switch colors so let's say that is one of the chords that defines the angle just like that and let's say that is the other chord that defines the angle just like that so how do we find the relationship between let's call this angle right here let's call it psy one how do we find the relationship between psy 1 and the central angle that subtends this same arc so I'm going to talk about the same arc that's that right there so the central angle that subtends the same arc will look like this it will look like this let's call that theta 1 what we can do is use what we just learned when one side of our of our inscribed angle is a diameter so let's construct that so let me draw a diameter here the result we want still is that this should be one half of this but let's prove it let's draw a diameter let's draw a straighter diameter than that so let's draw a diameter just like that and let me call this angle right here this angle right there let me call that let me call that sigh - let me call that sigh - and it is subtending this angle this arc right there let me do that in a darker color it is subtending this arc right there and so the central angle that subtends that same arc let me call that theta - now we know from the last from really just earlier part of this video that psy - psy - is going to be equal to 1/2 theta - right they share the diameter is right there the diameter is one of the chords that forms the angle so sy2 is going to be equal to 1/2 theta - 1/2 theta - this is exactly what we've been doing in the last video right this is an inscribed angle one of its and one of the the chords that define it is sitting on the diameter so this is going to be 1/2 of this angle of the central angle that subtends the same arc now let's look at this larger angle this larger angle right here psy 1 + sigh - write that larger angle is psy 1 psy 1 + sigh - sigh - once again this subtends this entire arc right here so it's going to be when it has a diameter as one of the chords that defines this huge angle so this is going to be 1/2 of the central angle that subtends the same arc we're just using what we showed what we what we've already shown in this video so this is going to be equal to 1/2 of this huge central angle of theta 1 plus theta 2 plus theta 2 right so far we've just used everything that we've learned earlier in this video now we already know that they at psy 2 is equal to 1/2 theta 2 so let me make that substitution this is equal to that so we can say that psy 1 psy 1 is eat oh sorry psy 1 plus instead of site - I'll write 1/2 theta - 1/2 theta 2 is equal to 1/2 theta 1 1/2 theta 1 plus 1/2 theta 2 plus 1/2 theta 2 we can subtract 1/2 theta 2 from both sides and we get our result psy 1 is equal to 1/2 theta 1 and now we are done we've proven the situation that the inscribed angle is always one half of the central angle that subtends the same arc regardless of whether the center of the circle is inside of the is inside of the angle out outside of the angle whether we have a diameter on one side and so any situation can be constructed as you know any other angle can be constructed as a sum of any or all of these that we've already done but hopefully you found this useful now we can actually build on this result to do some more interesting geometry proofs