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# Tangents of circles problem (example 1)

CCSS.Math:

## Video transcript

so we're told that angle a is circumscribed about circle o so this is angle a right over here's this 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 there are segments that would be part of tangent lines so if we were to continue so for example that right over there that's a tangent that that line is tangent to the circle and let me see if I could so 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 and any quadrilateral the sum of the angles are going to add up to 360 degrees and if you wonder where that camp comes from well you could divide a quadrilateral into two triangles where the sum of all the interior angles of a triangle or 180 and since you have two of them 360 360 degrees so this is 90 to plus 90 plus 90 plus question mark is going to be equal to 360 degrees so let me write that down 90 to plus 90 plus 90 so plus 180 plus let's just call this X X degrees plus X they off 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 equal to 180 and if we subtract 92 from both sides we get X is equal to let's see 180 minus 90 would be 90 and then we're going to subtract two more so it's going to be is equal 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 oh we've seen that before in the last question and they said what is the measure of angle D so we want to find we want to find let me make sure I'm on the right layer we want to find the measure of that angle 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 cuadrado a be OC 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 and 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's 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 some 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 of the 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 because that that interior angle intercepts intercepts arc CB right over here that tells us that the measure of Arc CB is also 100 degrees and so if we're trying to find this angle that measure of angle D that's an inscribed angle that intercepts the same arc and we've seen in previous videos that that an inscribed angle that intercepts an arc is going to have half the arcs measure so this is a hundred degree measured arc then the measure of this angle right over here is going to be fifty degrees so the measure of angle D is fifty degrees and we are done