 This video is Practice Problems 1. We have two diagrams where we're going to do them separately and find the missing values. The first diagram shows four inscribed angles in the circle, and we're asked to find the measure of angle ACD. This here is the measure of, or this here is angle ACD, and we have a theorem that talks about inscribed angles. I'm going to highlight this angle here as well. Angle ABD, and our theorem states that if we have two separate inscribed angles of a circle, we have this inscribed angle, the red one, and the blue one. If we have two separate inscribed angles of a circle that intercept the same arc, then those angles are congruent. You notice that both ACD and ABD have their intercepted arcs right here, arc AD, and when that happens, we know the measures of those inscribed angles are going to be congruent. So if the measure of angle ABD is 36 degrees, then the measure of angle ACD is also 36 degrees because they share this intercepted arc AD. Our next problem is asking us to find the measure of arc AD, and that's this arc right here. We just talked about this inscribed angle intercepting that arc, and we know we have a theorem that if we have an inscribed angle, the intercepted arc is going to be twice the measure. So if angle ABD is 36 degrees, we're just going to double that, and this intercepted arc is going to be 72 degrees, twice the measure, and that will lead us right into the next problem, because if we know that this intercepted arc is 72 degrees, arc AD, then we can just set that equal to the 4x term and solve for x. So we're just going along and putting in values that we know, and x equals 18. The next diagram shows an inscribed triangle, so we should notice a few things about this diagram. This triangle, we say, is inscribed in the circle, and the hypotenuse of this triangle, or I should say one side of the triangle, is actually the same as the diameter of the triangle. This creates a semi-circle here. There's a theorem that states if an inscribed angle of a triangle intercepts a semi-circle, the angle is a right angle, and all that means is that if you have a triangle inscribed in a circle and one of the sides goes through the diameter, this is always going to be a right triangle, no matter where this point is on the circle. So we can go ahead and put our little right angle mark in there, and we know that angle ABC is going to be 90 degrees, and we can go ahead and put that in for the first question on problem two here. An inscribed triangle is always going to be a right triangle, so the measure of that angle is 90 degrees. The next problem is asking us to find the measure of the arc. Remember the measure is the angle measure of that arc, and we know a couple of things about this. We just said that this is the diameter of the circle, and that creates a semi-circle there. We know that a semi-circle is 180 degrees, and so we don't need to do much more than just saying that's 180 degrees. Another thing to notice is that angle ABC is an inscribed angle. We know that's 90 degrees, we just talked about that, and we know that the measure of an inscribed angle is always going to be half the measure of the intercepted arc. Since this angle is 90 degrees, the intercepted arc would have the measure of 180 degrees. And then the last piece of this is asking us to solve for x. We just said that this arc, arc AC is 180 degrees, so we can set up that equation. If arc AC is 180 degrees, that gives us our equation. 180 equals 3x plus 30, and we just need to solve for x. Any time you're asked to solve for x, you're just going to set that equation up based on the theorems given to us and what we know about central angles in an intercepted arc, so x equals 50.