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# Inscribed shapes: angle subtended by diameter

CCSS.Math:

## Video transcript

on circle Oh below so this is circle Oh segment s II is a diameter so this is s II I can actually let me color that in so this is the diameter that's what they're telling us that's SE and they say what is the measure of angle I s II so what we care about the measure of angle is e I se4 trying to figure out this angle right over there and 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 that the angles the 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 is 20 has a measure of 27 degrees if we could figure out this angle right over here this is what would be angle s ie then if we know two angles two interior angles of a triangle we can figure out the third and this one s ie 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 right over here is going to be 180 minus 61 so what is that going to be 180 minus 60 to be 120 and then minus one and BS is is 119 119 degrees and so this angle that we 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 that 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 measure of angle is e is 34 degrees now I mentioned that there is multiple ways that we could figure this out let me 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 IC is still the thing that we want to figure out another way that we could approach it is well we know we have some angles inscribed angles on this circle and we know that if an inscribed angle intercepts the 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 could also use that information if we look at a different so 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 29 degrees and then we could look at this larger triangle we could look at this larger triangle right over here to figure out this entire angle to figure out this entire angle if we know this entire angle you subtract 29 then you figure out angle is e and so this large ortho what I've depicted is this kind of this magenta this 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 SL e so this angle right over here is going to be 180 minus 61 minus 27 sorry not minus 61 minus 90 minus 90 it's 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 90 minus 27 is 63 63 degrees so this large one over here is 63 degrees and the smaller one is 29 degrees and so angle is e 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 way 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 you know here we use this idea of an inscribed angle that intercepts a diameter and if you say hey how do we know I mean we prove it in other videos but it comes straight out of the idea that the inscribed angle the measure of an inscribed angle is going to be half of the of the measure of the arc that it intercepts and notice it's intercepting an arc that has a measure of 180 degrees its intercepting an arc that has a measure of 180 degrees and so this angle is going to be half of that since it's an inscribed angle not a central angle and so it's going to be a 90-degree angle