Inscribed shapes problem solving
- [Voiceover] What I wanna do in this video is 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. All right, 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 the measure of arc, I'ma write it 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 purple arc 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. That plus arc, WL, they are going to add up to 360 degrees. Let me write that down. So, the measure of arc, let's see, and this is going to be a major arc right over here. This is so LIW, the measure of arc LIW plus the measure of arc WL 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 WL is 90 degrees. So, if you subtract 90 degrees from both sides, you get that the measure of this large arc right over here, measure of arc LIW 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 in 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, the angle's 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 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, are 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 would 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 would 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.