Tangents of circles problem (example 1)

- [Voiceover] So we're told that angle A is circumscribed about circle O. So this is angle A right over here, we're talking about this angle right over there. And when they say it's circumscribed about circle O that means that the two sides of the angle they're segments that would be part of tangent lines, so if we were to continue, so for example that right over there, that line is tangent to the circle and (mumbles) and this line is also tangent to the circle. So you see that the sides of angle A are parts of those tangents, and points B and point C are where those tangents actually sit on the circle. So given all of that they're asking us what is the measure of angle A, so we're trying to figure this out right here and I encourage you to pause the video and figure it out on your own. So the key insight here and there's multiple ways that you could approach this is to realize, that a radius is going to be perpendicular to a tangent or a radius that intersects a tangent is going to be perpendicular to it. So let me label this, so this is going to be a right angle and this is going to be a right angle. In any quadrilateral, the sum of the angles are going to add up to 360 degrees, and if you wonder where that comes from well you can divide a quadrilateral into two triangles, where the sum of all the interior angles of a triangle are 180 and since you have two of them, it's 360 degrees. So this is 92 + 90 + 90 + question mark is going to be equal to 360 degrees. So let me write that down. 92 plus 90 plus 90 so plus 180. Plus, let's just call this X. X degrees. Plus X, they all have to add up to be 360 degrees so let's see, we could subtract 180 from both sides and so, if we do that we would have 92 plus X is equals to 180 and if we subtract 92 from both sides we get X is equals to let's see 180 minus 90 would be 90 and then we're just going to subtract two more so it's going to be X is equals to 88. So the measure of angle A is 88 degrees. Now let's do one more of these. These are surprisingly fun. All right. So it says, angle A is circumscribed about circle O, we have seen that before in the last question and they said what is the measure of angle D. So we want to find, let me make sure I'm on the right layer. We want to find the measure of that angle and let's call that, let's call that x again. So what can we figure out? Well just like in the last question we have a quadrilateral here, quadrilateral A, B, O, C and we know two of the angles, we know this is going to be a right angle we have a radius intersecting with a tangent, or part of a tangent I guess you could say. And then this would be a right angle and so by the same logic as we saw in the last question, this angle, plus this angle, plus this angle, plus the central angle are going to add up to 360 degrees. So let's call the measure of the central angle let's call that Y over there. So we have Y plus 80 degrees or we'll just assume everything is in degrees, so Y plus 80 plus 90 plus 90, so I could say plus another 180 is going to be equal to 360 degrees sum of the interior angles of a quadrilateral. And so let's see you have, we could have Y plus 80 is equal to 180, if I just subtract 180 from both sides if I subtract 80 from both sides, we get Y is equal to 100 or the measure of this is 100, the measure of this interior angle right over here is 100 degrees, which also tells us that the measure of this arc cause that interior angle intercepts... intercepts arc C, B, right over here, that tells us that the measure of arc C, B, is also 100 degrees. And so if we're trying to find this angle, the measure of angle D, that's an inscribed angle that intercepts the same arc. And we've seen in previous videos that an inscribed angle that intercepts that arc is going to have half the arc's measure. So if this is a 100 degree measured arc, then the measure of this angle right over here is going to be 50 degrees. So the measure of angle D is 50 degrees. And we are done.