Circles Unit Angles Inscribed angles and outside a circle

Hey, welcome back. We're doing circles part one lesson 8 angles inside and outside. Okay, I love these arcs and angles unit. It's one of my favorite things. Let's go ahead and jump in to your note taking guide. Okay, we have already covered these first two types of angles in earlier in the unit, but let's review them so we can have everything in one spot so you can really identify them. Okay, remember the words Waldo, while where's Waldo? Remember, tells us to look where is the vertex of the angle of the angle in relation to the circle. So this one has the vertex at the center, and we know that the angle and the arc have the same measure. So if this arc or this angle is a 100° and the arc measures a 100°. So let's do an example. So here's our angle. Our hair is our circle. And here's an angle, and let's say the angle measures 60, then we know the arc also measures 60. They are the same measure. Now let's look at the second case, which is the inscribed angle. Angles that are on the circle on, meaning the vertex is on the circle. Okay? The angle is half the measure of the arc. So you're going to take half the measure of the arc. Equals the angle. Okay, so let's do a quick example and remember these are equal. Okay. So if the arc measures 70 8, we would take half of that and half of that would be 30 9. So the angle is half of the arc. Or given the angle, let's say, is 80, then the arc up here is one 60. So that's the relationship, both forward and backward. Okay, now we're going to look at our two new cases. What happens if we have an angle that is in the circle, but it's not at the center, okay? I kind of call this a nomadic angle. It's wandering somewhere in the circle, and it's not in the center. So we can't use that the central angle one from up here at the top. We can't use that one. Okay, what we're going to do is we're going to look at the angles that are inside the circle. The angle measure is the average of the measure of the intercepted arcs. By the angle and its vertical angles. And that sounds kind of confusing, doesn't it? Okay, well not really once we get it drawn. Okay, let's say we're looking for this angle right here. Okay? All we have to do is look at kind of the arc up here in the arc down here. I kind of think of it as a bow tie, so I'm going to turn this first second. And kind of look at it as a bow tie rule. Okay, if this is your bow tie, we know that these two angles are congruent because they're vertical angles. So what arcs do they intercept this arc in this arc. So you've got arc one and R two. So what the rule says is we're going to average them. That means add and divide by two. So we're going to take arc, one plus arc two, and we're going to divide it by two, and that's going to equal the measure of the angle. In the center. So let's throw some numbers up there. So you can kind of see how this works. Let's say that the angle up here is a hundred. And the angle down here is a 120. To find this angle in the middle, or these two angles, I'm going to just take 100 plus one 20 and divide them by two, so the sum of those is two 20 and two 20 divided by two is 110. So that means each one of these angles measures a 110. It's the average of those. Okay, now remember, we're going to have to do another step here at some point, not necessarily yet. But I just wanted to preface this. What do we know about these angles? Well, if this one is a 110 and we have a straight line here, aren't these two supplementary. So if this is one ten, I can subtract it from one 80 and get this side. So this side would be 70 and we know these are vertical so that side would be 70. So anyway, just to preface that before we go on because you're going to be solving some of those. But the main concept here is to find these angles you add and divide by two. Okay. Angles outside the circle. Okay, how does that work? Well, you have an angle outside the circle and notice we have two secants. While secant crosses the circle twice, doesn't it? Now this one's drawn with two secants, but you can actually have it with two tangents or a tangent in a secant also. Okay, but the main thing is the vertex of the angle lies outside the circle. So the angle measure is half the difference of the measure of the arc center intercepted by the angle. So notice we have two arcs that are inside the mouth. We have this one, a small one, and this one, a larger one. We're going to take the different of those, which means we're going to subtract and divide by two. So we're going to take arc one, the larger arc minus arc to this smaller arc and divide them by two. And that gives us the measure of the angle outside the circle. Let's throw some numbers in there this time. Okay, let's say the arc one is one 20. And this little arc down here is 40. All we're going to do is take one 20 -40 and divide by two. So one 20 -40 is 80. And 80 divided by two is 40. So that means this angle out here is 40°. Now it's not because this arc is 40. It's because when we take the difference and divide by two, we get and divide by two, we get 40. Okay? So that's how you find angles on the outside of the circle. Now we're going to be going doing some forward and backwards once, but always stick with these formulas. Okay, I'm going to give you a helpful hand to help you remember when to add and when to subtract. So let me zoom out just a little. Okay, there's a very lopsided circle. There we go. That one's a little bit better. Okay, let's think about the bow tie rule and the one on the inside for a minute. Okay. And then we're going to look at the outside angle. Okay, first of all, what symbol do you see inside the circle? You see kind of if you look at it sideways, you kind of see it as a plus sign. So we're going to add the arcs on this one. And divide by two. What do you see as a symbol on the inside of this circle? It looks like just a straight line. So it's more like subtraction. So you're going to subtract this arcs. And divide by two. To find the angles. Okay? So there you go. Just a quick little visual that may help you remember the formulas for those. So the ones on the inside you add, the ones on the outside you subtract the arcs. Okay, so let's try a couple of these problems. And we've got a couple reviews, so let's look at these together. Let's see what we can do. Okay, second page of your note taking guide. Okay, let's look at this first one. Where is the vertex? The vertex is on the circle. So this is an inscribed angle. So all we need to do is take that and double it times two to get the arc. So 62 times two is one 24. So that is the arc measure from a to C so that's just an inscribed angle. How about this one? Second one. Okay, where's the vertex? The vertex again is on the circle, but notice this is a tangent. That's what makes it a little bit trickier to see, but you still use the same rule. All you're looking for is where is the vertex? The vertex is on the circle. So here is the angle. The arc is 84, so I'm at this time I'm going backwards. So I've got to make it smaller. So I'm going to take going to take 84 and divide by two, and that's 42. So the angle is 42. Degrees. Okay? So those are two inscribed angles. Now let's look at some of the new ones. Okay? We're looking for this angle right here. And it's not at the center of the circle. If it was, we would have a 113° angle and we'd be done. But it's not. So we have to use the bow tie roll. So let's kind of draw. And see which arcs we're using. We're using the 93 and the one 13. So since it's inside the circle, we're going to add the arcs. One 13 plus 93. Then divide by two, always remember to divide by two. So one 13 and 93 is what? Two ten. Two, 16. I'm grabbing my calculator because I don't want to make a silly mistake on my video. Let's see. Two O 6. Two O 6 divided by two is one O three. So that means the angle is going to be a 103°. And we know that the vertical angle on this side is also what? A 103. And if we needed to find it supplement, we could find these if we needed to because we know these two add to one 80. Okay. Let's go to this one. We're looking for a, D, let's look at the bow tie roll here. Okay. This time we're given the inside angle. And we've got to find the arc. Um. Remember, we add these two arcs together, divide by two to get the inside angle. So that means I'm going to have to call ADX because I don't know that. So let's take the bow tie roll X plus 84, all divided by two equals our central angle, 92, or our angle in the middle of the circle is not central so don't quote me on that. Okay. So let's solve that by just doing a cross product. I like to stick the 92 over one to do a cross product. So 92 times two is one 84. So let's jot that down. We got X plus 84 equals one. 84. Subtract 84. So X is 100, okay? So that means this side is 100 so that anchor is a hundred. Okay? So that's kind of how you would solve it if you're missing an arc. Just use your formula art plus arc divided by two is equal to the angle. Okay, so those are the inside the circle, but not at the center. Now let's look at these last couple, okay? This one's outside the circle. So I've got to look at my intercepted arcs, okay? Big arc, small art divided by two is equal to the angle. So 65 -15 and remember this is the subtraction one. So that's going to be what? 50 divided by two, so that's 25. So the angle out here is 25°. Big arc minus small arc divided by two. Remember, this is the subtraction one. Okay? Now here's a backwards one. We have the angle. We have one arc, but we're missing this arc. Use the same rule of thumb. Big arc, which is 70. Minus small arc. We don't know, so we're going to call that X all divided by two is equal to 27. My rule is cross products. So let's stick a one under to the 27 and do a cross product. 27 times two is 54. Is equal to 70 minus X so subtract 54 from 70 and we get what? 16 is equal to X so that means this arc is 16°. Okay? So that's how you work those out. Now it's your turn. You've got some practice problems on inside and outside. Good luck to you.