Tangents of circles problem (example 2)

angle a is a circumscribed angle on circle o so this is angle a right over here then when they say it's a circumscribed angle that means that the two sides of the angle are tangent to the circle so AC is tangent to the circle at Point C a B is tangent to the circle at point B what is the measure of angle a now I encourage you to pause the video now and to try this out on your own and I'll give you a hint a hint it will leverage the fact that this is a circumscribed angle as you could have as you could imagine so I'm assuming you've given a go at it so the other piece of information they give us is that angle D which is an inscribed angle is 48 degrees and it it intercepts the same arc so this is the arc that intersects a guess you could call it it intercepts this arc right over here it's the inscribed angle the central angle that inscribes that that intersects that same arc is going to be twice the inscribed angle so let's this is going to be 96 degrees I could put three markers here just because we've already used a double marker the notice they both intercept arc CB so you could you could some people say the measure of Arc CB is 96 degrees the central angle is 96 degrees the inscribed angle is going to be half of that 48 degrees so how does this help us well a key clue is that angle is a circumscribed angle so that means AC and a B are each tangent to the circle well a line that is tangent to the circle is going to be perpendicular to the radius of the circle that intersects the circle at the same point so this is this right over here this right over here is going to be a 90-degree angle and this right over here is going to be a 90-degree angle OC is perpendicular to C a OB which is a radius is perpendicular to B a which is a tangent line and they both intersect right over here at B now this might jump out at you we have a quadrilateral going on here a B OC is a quadrilateral so it's sides are going to add up its sides are going to add up to 360 degrees so we could know we know we could write it this way we could write the measure of angle a plus 90 degrees plus 90 degrees plus another 90 degrees plus another 90 degrees plus 96 degrees plus 96 degrees is going to be equal to 360 degrees is going to be equal to 360 degrees or another way of thinking about it if we subtract 180 from both sides so if we subtract that from both sides we get the measure of angle a plus 96 degrees plus 96 degrees is going to be equal to is going to be equal to 180 degrees or another way of thinking about it is the measure of angle a or that angle a and angle oh right over here you could call it angle C Oh B that these are going to be supplementary angles that they add up to 180 degrees so if we subtract 96 degrees from both sides we get the measure of angle a measure of angle a is equal to I don't wanna make that look like a less than symbol let me make it measure of angle this one actually looks more like a measure of angle a is equal to 180 minus 96 to see 180 minus 90 would be 90 and then we subtract another 6 gets us to 84 degrees