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# Inscribed quadrilaterals proof

Video transcript

- [Voiceover] So I have a
arbitrary inscribed quadrilateral in this circle and what I wanna prove is that for any inscribed quadrilateral, that opposite angles are supplementary. So when I say they're supplementary, 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 gonna prove
it is we're gonna 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, then we've proven that opposite angles for an arbitrary quadrilateral
that's inscribed in a circle are supplementary, 'cause
if 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 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 an inscribed angle, the vertex of this angle
sits on the circle, the relationship between
and inscribed angle 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 if this angle measure is 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 degrees. 360's all the way around. The blue is all the way
around minus 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 arc right over here? Well, it's this angle. It's the angle that we
wanted to figure out in terms of x. Wow, I'm having trouble changing colors. It is that angle right over there. Notice, the two sides of this angle, they intercept, this
angle intercepts that arc. So, once again, the measure
of an inscribed angle is gonna half the measure of the arc that it intercepts. So what's 1/2, what is 1/2 times 360 minus 2x? Well, one 1/2 times 360 is 180. 1/2 times 2x is x. So the measure of this angle is gonna 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.