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# Solving inscribed quadrilaterals

CCSS.Math:

## Video transcript

what I want to do in this video see if we can find the measure of angle D if we could find the measure of angle D and like always pause this video and see if you can figure it out and I'll give you a little bit of a hint it'll involve thinking about how an inscribed angle relates to the corresponding to the measure of the arc that it intercepts so think about it like that alright so let's work on this a little bit so what do we know what do we know well angle D angle D intercepts an arc it intercepts this fairly large arc that I'm going to highlight right now in this purple color so it intercepts that arc we don't know the measure of that arc or at least we don't know the measure of that arc yet if we did know the measure of this arc that I'm highlighting then we know that the measure of angle D would just be half that because the measure of an inscribed angle is half the measure of the arc that it intercepts we've seen that multiple times so if we knew the measure of this arc we would be able to figure out what the measure of angle D is but we do know we don't know the measure of that arc but we do know the measure of another arc we do know the measure of the arc that completes the circle so we do know the measure of this arc you might be saying hey wait how do we know that measure it's not labeled well the reason why we know the measure of this arc that I've just highlighted in this teal color is because the inscribed angle that intercepts it they gave us the information they said this is a 45 degree angle so this is a 45 degree angle then this over here is a 90 degree 90 degree arc the measure of this arc is 90 degrees the measure of Arc I guess you could say this is a measure of Arc I'm ready this way the measure of Arc WL WL is equal to 90 degrees it's twice that the inscribed angle that intercepts it now why is that helpful well if you go all the way around the circle you're 360 degrees so this this purple Ark that we cared about that we said hey if we could figure out the measure of that we're gonna be able to figure out the measure of angle D and that Plus met arc WL they are going to add up to 360 degrees let me write that down so the measure the measure of Arc let's see and this is going to be this is going to be a major arc right over here this is so L I W the measure of Arc L I W the measure of Arc W L plus the measure of Arc WL Plus this right over here that's going to be equal to 360 degrees this is going to be equal to 360 degrees now we already know that this is 90 degrees we already know W L is 90 degrees so if you subtract 90 degrees from both sides you get that the measure of this large arc right over here a measure of Arc L i w is going to be equal to 270 degrees I just took 300 I went all the way around the circle I subtracted out this 90 degrees and I'm left with 270 degrees so let me write that down this is the measure of this arc and purple is 270 degrees and now we can figure out the measure of angle D it's an inscribed angle that intercepts that arc so it's going to have half the measure of the angles going to have half the measure so half of 270 is 135 degrees and we're done and you might notice something interesting that if you add 135 degrees plus 45 degrees that they add up to 180 degrees so it looks like at least for this case that these angles these opposite angles of this inscribed quadrilateral it looks like they are it looks like they are supplementary so an interesting question is are they always going to be supplementary if you have a quadrilateral an arbitrary quadrilateral inscribed in a circle so each of the vertices of the quadrilateral sit on the circle if you have that our opposite angles of that quadrilateral are they always supplementary do they always add up to 180 degrees so I encourage you to think about that and even prove it if you get a chance and the proof is very close to what we just did here in order to prove it you just have to do it with more general numbers like you know instead of saying 45 degrees you could call this X and then you would want to prove that this right over here is wouldn't have to be 180 minus X so I encourage you to do that on your own but I'm gonna do it in a video as well so you can check if our reasoning is similar