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Inscribed shapes: angle subtended by diameter

Video transcript
- [Voiceover] "On circle O below", so this is circle O, "segment SE is a diameter." So this is SE, let me color that in. So this is a diameter, that's what they're telling us. And they say, "What is the measure of "angle ISE?" So what we care about, the measure of angle ISE. I, S, E. So we're trying to figure out this angle right over there. Now like always, I encourage you to pause the video and see if you can work through it yourself. So there's a bunch of ways that we can actually tackle this problem. The first one that jumps out at me is there's a bunch of triangles here, and we can use the fact that the angles, interior angles of a triangle add up to 180 degrees. So we could look at this triangle right over here. And so we know one of the angles already. We know this angle has a measurement of 27 degrees. If we could figure out this angle right over here, this would be angle SIE, then if we know two interior angles of a triangle, we can figure out the third. And this one, SIE, we can figure it out because it's supplementary to this 61-degree angle. So this angle right over here is just going to be this angle plus the 61-degree angle is going to be equal to 180 degrees, because they are supplementary. Or we could say that this angle over here is going to be 180 minus 61. So what is that going to be? 180 minus 60 would be 120, and then minus one gives us 119. 119 degrees. And so this angle that we are trying to figure out, this angle plus the 119 degrees, plus the 27 degrees, is going to be equal to 180 degrees. Or we could say this angle is going to be 180 minus 119, minus 27. Which is going to be equal to, so let's see, 180 minus 119 is 61. And then 61 minus 27 is going to be 34. So there you have it. The measure of angle ISE is 34 degrees. Now I mentioned that there is multiple ways that we could figure this out. Let me do it one more way. So let me unwind everything that I just wrote. We already figured out the answer, but I want to show you that there's multiple ways that we can tackle this. So ISE is still the thing that we want to figure out. Another way that we could approach it is we know we have some angles inscribed angles on this circle, and we know that if an inscribed angle intercepts a diameter, then it's going to be a right angle. It's going to be a 90 degree angle. So this angle right over here is a 90 degree angle. And we can use that information to figure out this angle, and we can also use that information if we look at this triangle, we could use 90 plus 61 plus this angle is going to be equal to 180 degrees. So this angle right over here, another way to think about it, it's going to be 180 minus 90 minus 61. Which is equal to, 180 minus 90 is 90, minus 61 is 29 degrees. So this one right over here is 29 degrees. And then we could look at this larger triangle. To figure out this entire angle. If we know this entire angle, you subtract 29 then you figure out angle ISE. And so this large, or what I've depicted, this kind of magenta measure right over here of that angle, plus 90 degrees, plus 27 degrees, is going to be equal to 180 because they're the interior angles of triangle SLE. So this angle right over here is going to be 180 minus 61 minus 27. Sorry, not minus 61, minus 90. It's 180 minus 90, minus 27 is going to give us this angle right over here because the three angles add up to 180. So minus 90, minus 27. Which is equal to, so 180 minus 90 is 90. 90 minus 27 is 63. 63 degrees. So this large one over here is 63 degrees, and then the smaller one is 29 degrees. And so angle ISE, which we set out to figure out, is going to be 63 degrees minus the 29 degrees. So 63 minus 29 is once again equal to 34 degrees. So the way I did it just now, a little bit harder. It really depends what jumps out at you. The first way I tackled it, it does seem a little bit easier, a little bit clearer. But it's good to see these different things. And at least here, we used this idea of an inscribed angle that intercepts a diameter. And if you say, "Hey, how do we know?" Well, we've proven in other videos, but it comes straight out of the idea that the measure of an inscribed angle is going to be half of the measure of the arc that it intercepts. And notice, it's intercepting an arc that has a measure of 180 degrees. It's intercepting an arc that has a measure of 180 degrees. And so this angle's going to be half of that since it's an inscribed angle, not a central angle. So it's going to be a 90 degree angle.