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# Inscribed quadrilaterals proof

## Video transcript

so I have a arbitrary inscribed quadrilateral in this circle and what I want to prove is that for any inscribed quadrilateral that opposite angles are supplementary so when I say they're supplementary this the measure of this angle plus the measure of this angle need to be 180 degrees the measure of this angle plus the measure of this angle need to be 180 degrees and the way I'm going to prove it is we're going to assume that this the measure of this angle right over here that this is X degrees and so from that if we can prove that the measure of this opposite angle is 180 minus X degrees that we've proven that opposite angles for an arbitrary quadrilateral that's inscribed in a circle are supplementary because this is 180 minus X 180 minus X plus X is going to be 180 degrees so I encourage you to pause the video and see if you can do that proof and I'll give you a little bit of a hint it's going to involve the measure of the arcs that the various angles intercept so let's think about it a little bit this this this this angle that has a measure of x degrees it intercepts this arc so we see one side of the angle goes and intercepts the circle there the other side right over there and so the arc that it intercepts I am highlighting in yellow I am highlighting it in yellow trying to color it in so there you go not a great job at coloring it in but you get the point that's the arc that it intercepts and we've already learned in previous videos that the relationship between the between an inscribed angle this the the vertex of this angle sits on the circle the relationship between an inscribed angle and then the arc that I did and the measure of the arc that it intercepts is that the measure of the inscribed angle is half the measure of the arc that it intercepts so this has this angle measures X degrees then the measure of this arc is going to be 2x 2x degrees all right well that's kind of interesting but let's keep going if the measure of that arc is 2x degrees what is the measure of this arc right over here the arc that completes the circle well if you go all the way around the circle that's 360 degrees so this blue arc that I'm showing you right now that's going to have a measure of 360 minus 2x minus 2x minus 2x degrees 360s all the way around the blue one is all the way around - the the yellow arc what you have left over if you subtract out the yellow arc is you have this blue arc now what's the angle that intercepts this blue arc what's the inscribed angle that intercepts this blue this blue arc right over here what's this angle it's the angle that we wanted we wanted to figure out in terms of X it is it is I'm having trouble changing colors it is that angle right over there notice the two sides of this angle they they intercept this angle intercepts that arc so once again the measure of an inscribed angle is going to be half the measure of the arc that it intercepts so what's 1/2 what is 1/2 times 360 minus 2x well 1/2 times 360 is 180 1/2 times 2x is X so the measure of this angle is going to be 180 minus X degrees 180 minus X degrees and just like that we've proven that these opposite sides for this arbitrary inscribed quadrilateral that they are supplementary you add these together X plus 180 minus X you're going to get 180 degrees so they are supplementary