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PROVING A THEOREM Use the diagram below to prove the Angles Outside the Circle Theorem (Theorem 10.16 ) for the case of a tangent and a secant. Then copy the diagrams for the other two cases on page 563 and draw appropriate auxiliary segments. Use your diagrams to prove each case.
So we're back with Maura about circles and the special angles that are formed and how toe measure or calculate those angles. Now, when I say special angles, I mean special angles formed by certain line segments that cross through the circle. So a few background concepts first, here's your circle. Recall that if you have a circle and you have a segment that has it's end points on the circle, we call this accord and that you can have minor chords, which is this type of cord or major chord, which is a line say, when that goes through the center. Both of these records, however, the one I'm pointing out is a minor chord, the one that goes to the diameter. That part of me that the one that is the diameter that goes to the center of the circle is called a major chord. Well, sometimes you can have cords cross like so, but not go through the center so we would know how to find the measure of angles and arcs If we knew that they went through the center nicely because we could use the rules for how to find the measurements of a central angle or inscribed angle. That's not the case if our cords cross it, a point that is not the center. So now we have some angles in the interior of the circle that don't abide by those two rules. So here's a couple of themes for these unique angle situations. If you have two cords crossing, then the angle formed by the two intersecting lines is going to be half of the some of the two arcs that they sub tend. So this arc plus this arc divided by two, will equal the angle in question. So an example would be like this. Here's a circle. Here's my center, but here's my two chords. Let's say I knew that this ark was 20 degrees in this arc was 60 degrees completely making this up. Therefore, this angle here is going to be half the sum of the two arcs That's gonna be half of 80 or, in other words, 40 degrees. So this angle here would equal 40 degrees. Okay, that's one of three therms. I'm gonna show you. Okay, if we had something very similar to that, but we had the Okay, the cords extend from the circle. Pardon me. So let's say we had this going on now. First of all, what I'm drawing in green. That's accord. That's accord if you extend it and look at the bigger picture. Really, What we're dealing with is these lines called sequence. Both of these are known as sequence S E C. A N T sequence sequence are lines. Raise land segments that cross a circle twice, so notice that each of these rays is crossing the circle twice. Now what you get is an angle on the exterior of the circle. But that angle, like the previous example sub, tends to works of the circle. This time, the angle in question is going to be one half the difference of the two. So let's say this were 80 and this were 20 degrees, and we were given. Those arcs are 80 and 20 degrees, respectively. Then the angle excess in question would be one half, and then it would be 80. Pardon me, minus 20 which would be one half of 60 or, in other words, 30 degrees. So if the angle were formed by two chords or sequence with inside of the circle, then the angle is going to be essentially the average of the two arcs that its substance one R plus the other arc divided by two. If the angle is formed in the outside of the circle, as you can see here, then the angle is going to be equal to the half of the difference of the two. Okay, now it's time to introduce one new type of segment. We'll get to know a lot more in a few lessons from now. But if you have a circle and you have a line that goes in a liner, a line segment or array that just touches the circle, do a little better just touches the circle on exactly one point. This segment is known as a tangent, so it's possible that we could have something like this where we have one second and we have one tangent and we have this angle here. So if we wanted to figure out the Rule two number for how to find this angle, X, let me draw that in different color. Then, once again like the previous example, it's going to be one half this ark minus disorder. Yeah, because we still have an angle that sub tending two different arcs of the circle. But there's a very special situation. What if we had a circle? Here's the center we had a tangent, but this time the angle came from what we call the point of tangent C that is called the point of Tangent C. What is this angle then? Well, since it on Lee sub tends one arc, technically, it sometimes, too. But one of them is zero. So instead of having one arc minus the other divided by two, So instead of finding half the difference you technically are, it's just the one arc zero because it only sub tends technically this particular arc. So let's say I knew that this arc here were 100 degrees thin. This angle in question is gonna be one half of 100 or, in other words, 50. So those were three additional angle the're ums or not doing the formal proofs by any means. But those air three additional rules that pertain to angles inside, outside and technically sort of on the circle, depending on how the sequence and tangents air interacting with each other is there different than the central and inscribed angle rules because we're not coming from the center, Nor are we coming from directly on the circle. Mawr examples coming soon