# US4 SECANTS AND TANGENTS

This eighth grade mathematics lesson focuses on the measurement of angles formed by secants and tangents intersecting with a circle. It is the fourth lesson in a six-lesson unit on this topic. The lesson is 45 minutes in duration. This is an advanced eighth grade geometry class. There are 15 students enrolled in the class.

Time | Caption |
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00:00:00 | We're doing nine point four today. |

00:00:01 | (inaudible) |

00:00:06 | Are you ready? |

00:00:07 | No. |

00:00:09 | If you need to sharpen your pencil, you better do that right this second. |

00:00:17 | (inaudible) homework? |

00:00:19 | Probably not there, Matt. Oh, goodness. |

00:00:22 | (inaudible) |

00:00:35 | I hope you guys have your activities ready from the boxes from nine point four. |

00:00:40 | That was the only other thing that I asked you to do. |

00:00:43 | I heard a couple of things- |

00:00:44 | Ben, you don't even have your books open, ready to go. What's going on? |

00:00:52 | I thought you were going to be the star of my show. |

00:00:54 | I'm definitely the star. |

00:01:00 | We're gonna go through the activities kind of quickly, 'cause you should have already had them done and ready to go. |

00:01:19 | Okay? Here's your problem of the day. |

00:01:23 | Ready to go. |

00:01:35 | A little review from nine point three. |

00:01:53 | You may already have this in your notes. |

00:01:58 | You might have to look back, try it from memory or look in your notes. |

00:02:17 | I wanna know about the inscribed angle theorem, the right angle corollary, and the arc intercept corollary. |

00:02:53 | Page 582 if anybody's wondering. |

00:02:58 | :00] |

00:04:37 | Okay. |

00:04:39 | Help me out with the inscribed angle theorem. Tell me what- what it is. |

00:04:43 | In your own words, how would you describe that one? |

00:04:48 | Anybody? |

00:04:49 | Suzy? |

00:04:51 | If an angle's inscribed in a circle and it intercepts part of the circle, then the angle's measure is equal to half of the other angle. |

00:04:59 | Okay. It ends up coming out to be half of the measure of the arc that it intercepts. Right? |

00:05:05 | How about right angle corollary? |

00:05:07 | Matt. |

00:05:09 | If an inscribed angle intercepts a semicircle, then the angle is a right angle. |

00:05:13 | Okay. If it intercepts a semicircle, then angle turns out to be- |

00:05:17 | Right angle. |

00:05:18 | A right angle. Okay? |

00:05:20 | And arc intercept corollary, Margaret. |

00:05:22 | When you have two inscribed angles and they intercept the same arc (inaudible). |

00:05:28 | Okay. Good. When they intercept the same arc, then they have the same- |

00:05:33 | Measure. |

00:05:34 | Measure. Good. |

00:05:36 | Okay. |

00:05:37 | You can open to nine point four, page 588. |

00:05:44 | The first thing we're gonna look at is what kind of angles that we're gonna be talking about today. |

00:05:51 | They're angles formed by secants and tangents. Here's all different cases. |

00:06:08 | It all depends on where the vertex is... and the type of lines that it- that it includes. |

00:06:19 | If the vertex is actually on the circle, you have secant and a tangent right here, or two secants here. |

00:06:31 | The second group is where the vertex is inside the circle, but it's not on the center usually. |

00:06:36 | Okay. Talking about here. |

00:06:38 | Last- the last group, the vertex is outside the circle. You have two tangents, two secants over here or a secant and a tangent. Okay? |

00:06:46 | That's all gonna help you determine how to find the measure. |

00:06:51 | Are you ready for the activities? |

00:06:53 | Take that- out, whatever you brought from Friday for homework. |

00:07:09 | We're not gonna go over each one of these in deep detail, 'cause you should have already had most of them done. |

00:07:14 | We'll go over the first, probably, row or two, and then we're gonna talk about the general pattern at the end with X. |

00:07:21 | So, first one. |

00:07:24 | You have this circle with secant and tangent. The secant and tangent angle is a right angle here. Okay? |

00:07:31 | The secant contains the center of the circle. What do you think the measure of A V C, angle A V C is? |

00:07:37 | What'd you come up with? Michaela? |

00:07:40 | Ninety degrees. |

00:07:41 | Why do you say 90 degrees? |

00:07:43 | Because it says that secant and tangent angle is a right angle and it (inaudible). |

00:07:51 | Okay. So- and the measure of a semicircle here, arc A V is what? |

00:07:55 | A hundred and eighty degrees. |

00:07:57 | A hundred and eighty degrees. |

00:07:58 | So what can you get from that? If the whole arc is 180 and the angle is 90, what does that kind of remind you of? |

00:08:05 | (inaudible) |

00:08:07 | What's that? |

00:08:08 | The right angle corollary. |

00:08:09 | Okay. Yes. Kind of like the right angle one or where- basically because it's half. Right? |

00:08:18 | It's 180 degrees and this is 90, you can come up with this checkpoint here that says the relationship between |

00:08:26 | the inscribed angle and the intercepted arc is the fact that it's half of what that arc is. |

00:08:33 | That was the easy part. |

00:08:36 | Okay. Look at number two. The secant tangent angle- we're gonna go over- this was the- if it's the tangent and it's a right angle. |

00:08:43 | Next we're talking about the acute angles. What happens if the angle's acute? It's not a straight. |

00:08:48 | It's not coming straight through the center, so it's gonna be a little more tricky for you. |

00:08:54 | Look at arc A V. |

00:08:57 | It tells you that it's 120 degrees and then it gives you 120 degrees for angle one, here. |

00:09:05 | How do you know that? How can- how do they justify that? Leah? |

00:09:10 | Because it's the central angle, and the central, and the arc that they intercept equals the measure. |

00:09:15 | Good. That's back to the beginning of the chapter where the central angle is gonna be the same as whatever that arc that it intercepts. |

00:09:21 | Correct? |

00:09:23 | Angle two. Thirty degrees. Where do they come up with that one? |

00:09:27 | Go ahead. |

00:09:29 | Triangle A P V is isosceles, so angle three and angle two have to be congruent. |

00:09:36 | Okay. |

00:09:37 | And since all angles in a triangle have to add up to 180 degrees, you take 180- |

00:09:43 | You subtract 120 and you divide them into 65. |

00:09:46 | Very good. You get 30 degrees. |

00:09:50 | So where do we come up with this one? P V C. P V C. |

00:10:00 | That's the one they left blank in this first- first box. |

00:10:10 | You're on a roll, let's go. |

00:10:12 | Yeah, P V C has to be a right angle because it intercepts V C. |

00:10:19 | Okay. They say radius to that tangent we learned that- |

00:10:22 | Oh, yeah. |

00:10:24 | Back in the beginning, nine two or nine three. It had to be nine two because we just went over nine three. |

00:10:31 | Nine two, that- the radius that intercepts that has that same tangent point... that's gonna be 90 degrees. |

00:10:41 | That wasn't the hard part. |

00:10:43 | Measure of angle A V C. How would you get that one? |

00:10:49 | A V C. They say 60 degrees. |

00:10:54 | Go ahead. |

00:10:55 | If A V C is 90 degrees and angle two is 30 degrees (inaudible). |

00:11:02 | Okay. And you get 60. |

00:11:03 | This kind of- this being 60 degrees, how does that relate to back here, the arc? |

00:11:09 | Half. |

00:11:10 | Matt says it's half. Mike says it's half. |

00:11:14 | Okay. So the measure of an acute secant tangent angle with its vertex on the circle is- |

00:11:20 | Half. |

00:11:21 | One half the measure on the intercepted arc. |

00:11:25 | So we got two cases down. We got- if it's a 90 degrees angle with a secant and tangent, we got acute angles. |

00:11:35 | What do you think this... general statement... here, measure of arc A V- if it's X, what's angle one gonna be? |

00:11:48 | X. |

00:11:49 | X. |

00:11:50 | Okay. What about the measure of angle two? Michaela, describe that for us. |

00:11:57 | Suzy? |

00:11:59 | You first have to do 180 minus X (inaudible) and then you divide it by two. |

00:12:05 | Divide it by two. |

00:12:06 | Then they'll be the same as three. |

00:12:07 | These angles are gonna be the same. |

00:12:09 | Okay? |

00:12:10 | How about P V C? That's going to be- |

00:12:16 | Ninety degrees. |

00:12:17 | Ninety degrees. Okay. That's not gonna- that's not gonna change. |

00:12:22 | And so the last here- this angle is going to be what? |

00:12:25 | Half of the arc. |

00:12:27 | Which is? |

00:12:29 | Half X. |

00:12:30 | X. Okay. |

00:12:34 | So we know it works for that same intersected arc- intercepted arc theorem, works for 90 degrees, works for acute angles. |

00:12:45 | Last case is obtuse angles. |

00:12:49 | This one is a little bit more tricky. You might have a little bit more trouble doing this one. |

00:12:56 | If it's obtuse, they give you A X V around this way, is 200 degrees- |

00:13:02 | And they come up with this measure of angle one being 160. |

00:13:05 | How do you think they got that? |

00:13:13 | Melissa? |

00:13:15 | Full circle has to equal 360 degrees, take away the 200. |

00:13:20 | Good. If you add that whole circle it's 360, then the central angle on the other side- |

00:13:25 | We take away 200, it's gonna leave you with 160. |

00:13:28 | Okay. So that gives you angle one. |

00:13:30 | Measure of angle two here. How do you get that one? |

00:13:35 | Um, the triangle- |

00:13:39 | Triangle "um"... |

00:13:41 | Serum- theorem. It has to be equal to 180. |

00:13:42 | Okay. Equal 180, so- |

00:13:44 | If angle one is already 160, then angle two and angle three are equal. |

00:13:48 | Good. That leaves you with 20, which means one of them has to be 10. |

00:13:53 | Okay. How about P V C? |

00:13:56 | Anything different about that one than the other example? |

00:13:58 | No. |

00:13:59 | No. It's gotta be 90. |

00:14:00 | I'm just gonna copy that one down right now. |

00:14:02 | And how about the last one, this whole A V C. The actual obtuse angle. It says it's 100. |

00:14:11 | If you're following a pattern, what are you getting? |

00:14:13 | Half. |

00:14:14 | It's gonna be- |

00:14:15 | Half. |

00:14:16 | Half. |

00:14:17 | Okay? So, I'm just gonna fill this in now. |

00:14:19 | Measure of an obtuse secant tangent angle with its vertex on a circle is one half the measure of its intercepted arc. |

00:14:29 | So let's fill this in. |

00:14:32 | For measure of angle A V C, how would you relate that back to X for any- for any measure? |

00:14:36 | X over two. |

00:14:37 | X over two. |

00:14:39 | Okay. I'm just gonna work my way backwards. We know the 90. |

00:14:43 | Well, let's go to angle one. |

00:14:44 | How would you figure out angle one if you know X for A X V? |

00:14:48 | Three sixty minus X. |

00:14:49 | Three sixty minus X. |

00:14:53 | Oh, I was gonna write X over two. Three sixty minus X. |

00:14:59 | And Katie, how would you do that last one, that angle two? |

00:15:03 | Angle two would be (inaudible). |

00:15:09 | This one's a little bit tricky. |

00:15:11 | The 180 thing works like- I know how to explain it- I'm not sure exactly what the equation is, but I could do it. |

00:15:18 | Okay. |

00:15:19 | Like I- |

00:15:20 | So tell me what to do and we'll try to put it in X for you. |

00:15:23 | All right. You have to get the measure of angle A P V. |

00:15:30 | Okay. |

00:15:31 | And then- |

00:15:32 | That's 360 minus X. |

00:15:33 | Yeah. |

00:15:34 | Okay. |

00:15:35 | And then you would take the other two since it's an isosceles triangle and- |

00:15:39 | (inaudible) have to be equal so you have divide whatever the original angle is. |

00:15:44 | Like it- I don't know. I'm sorry. |

00:15:47 | You subtract A P V from 180 and then divide it by two. |

00:15:54 | That's what I would do anyway. |

00:15:55 | Okay. |

00:15:56 | So, 180 minus what we get from angle one. I'm gonna put measure of angle one right here. |

00:16:01 | Minus measure of angle one, and then you take that and divide it by... two. |

00:16:06 | Two. |

00:16:07 | Good. Sorry. |

00:16:10 | (inaudible) |

00:16:12 | I couldn't fit all that in that little, little box. |

00:16:15 | So you get the theorem. This kind of- all these build upon each other, you get the 90, the acute, and the obtuse angles. |

00:16:22 | They all come out to this theorem, where no matter what, if you have that secant and the tangent- |

00:16:28 | It could be a chord and a tangent, you're gonna come up with what? |

00:16:33 | Nine. |

00:16:34 | If you have an angle that intersects with a secant and a tangent all in the circle? |

00:16:39 | It's always gonna come up to be- |

00:16:40 | Half. |

00:16:41 | Half of that arc. |

00:16:48 | You have any questions on that? |

00:16:51 | I'm gonna wait to give you an example for that one until we get to the other side. |

00:16:55 | Turn the page. |

00:17:06 | I guess I don't have one for this page, so we're gonna have to do this, just out loud together. |

00:17:13 | So look at the top for activity two. |

00:17:21 | They're kind of similar, but you have now where the vertex is inside the circle- |

00:17:28 | You have the intersection of two secants inside the circle, and they want to figure out how you can get those angle measures. |

00:17:37 | So look at the first one across for number- table two. |

00:17:42 | You have the measure of arc A C gives you 160. |

00:17:47 | Then you have B D. That gives you 40 degrees. |

00:17:50 | How about- where did they get measure of angle one, do you think? |

00:17:56 | It says 80 degrees. Where do you think they got that? |

00:18:01 | Leah? |

00:18:03 | Arc A C equals 160, and angle one is an inscribed angle, so it's half of A C. |

00:18:13 | Yes. Good. |

00:18:14 | And since A C is 160, half is 80. |

00:18:16 | Good. Good. |

00:18:20 | That's what we actually just talked about reviewing from nine point three. |

00:18:25 | So, how about angle two? |

00:18:28 | Melissa? |

00:18:29 | B D is 40 (inaudible), so angle of that, so it's half. |

00:18:37 | Okay. Good. |

00:18:39 | How about A V C? |

00:18:41 | It's 100 degrees. |

00:18:45 | Similar to what we were doing- been doing, but it's got a slightly different twist on it. |

00:18:48 | Where do you think they could have gotten the 100 degrees if you're looking at that circle? |

00:18:53 | A V C is the angle we're looking at. |

00:19:04 | We know arc A C and we know arc B D. |

00:19:09 | Where could they have gotten that 100 degrees? |

00:19:16 | Michaela? |

00:19:17 | There's two ways you can get it. The first way is to add the measures of angle one and angle two together. |

00:19:22 | And the second way is to add the measure of the two arcs together and divide it by two. |

00:19:26 | Okay. |

00:19:27 | So Michaela says there's two ways. |

00:19:28 | She thinks you can do 160 plus 40 and you get 200, divide it in half, and that gives you 100. |

00:19:36 | Or she got 80 plus 20 equals 100. |

00:19:42 | Okay? Do we have anything to back up either one of those more than the other? |

00:19:47 | Or do you want to just do the next row and see what we come up with? |

00:19:50 | Yeah. |

00:19:51 | Let's do the next row. So the next one says if A C is 180 degrees, B D is 70 degrees, what about angle one? |

00:20:04 | Or angle two? |

00:20:08 | Those are the inscribed angles. |

00:20:16 | If A C is 180, what does that angle one have to be? |

00:20:21 | Ninety. |

00:20:22 | Ninety degrees. Okay? Half. |

00:20:23 | How about B D? |

00:20:29 | Seventy degrees. Seventy degrees. What do you think? Inscribed angle. |

00:20:33 | Thirty-five. |

00:20:34 | Thirty-five. |

00:20:35 | So what does that leave us with for A V C? |

00:20:42 | With Michaela's first method, it would be- what did we come up with? Ninety and 35 added together. What would that give us? |

00:20:50 | One twenty-five. |

00:20:51 | One twenty-five. |

00:20:52 | What about the other way? If we do 180 plus 70, what does that give us? |

00:20:58 | Two fifty. |

00:20:59 | Two fifty. |

00:21:00 | Divide that. |

00:21:01 | Divide that in half, you get- |

00:21:02 | One twenty-five. |

00:21:03 | One twenty-five. They both work. |

00:21:06 | That's interesting. |

00:21:13 | What do you think? |

00:21:20 | First angle is half the A C. |

00:21:26 | Okay. What Matt says- |

00:21:29 | That for measure of angle one has to be half of the arc A C. |

00:21:34 | Okay. So you feel like we're talking about the general one again, he says, we're down to the last row with X- |

00:21:40 | Measure of angle one has to be half of A C. Do we agree with that? |

00:21:45 | That's inscribed angle. |

00:21:47 | How about measure of angle two has to be half of- |

00:21:52 | B D. |

00:21:53 | B D. X two. |

00:21:56 | So what do you think about measure of A V C? Katie? |

00:22:00 | What if you added the two angles together at the top and that was- whatever that is, if it's the triangle- |

00:22:06 | You get that other measure, and then it's an 80 degree thing. See if you subtract that from 180. Like- |

00:22:15 | Okay. Say that again. |

00:22:16 | And then it's on the 180 degrees line cause that's where the line is- |

00:22:23 | Okay |

00:22:24 | All right. That you add the two angles at the top, and subtract that from one- that from 180 because it makes a triangle. |

00:22:28 | So you subtract that from 180 and that gives you what the other angle is. |

00:22:33 | Okay. You're talking about A V C or D V B? |

00:22:36 | A V C. The first one. |

00:22:40 | And it works for the other one, too. |

00:22:42 | Okay. |

00:22:44 | Is that a lot to do for a general rule? |

00:22:47 | Maybe, but it works. |

00:22:48 | It works? Okay. |

00:22:50 | So now we have maybe three different ways to find it? |

00:22:53 | But that actually- that actually proves that- what Michaela said works. |

00:22:58 | 'Cause (inaudible). |

00:22:59 | Which one there? She did two. |

00:23:00 | Both of them. |

00:23:01 | Well both of them- |

00:23:02 | It proves that both of them work. |

00:23:03 | Both of them got the same thing. |

00:23:04 | Both of them got the same answer as the other one that Katie just said, and that really proves it. |

00:23:08 | The 125 and the 100 that we already went over? |

00:23:11 | Yeah. |

00:23:13 | Okay. |

00:23:14 | Because basically just adding angle one and angle two together is simplifying adding A C and B D- B D together and divide it by two. |

00:23:22 | 'Cause if you do 180- |

00:23:24 | Okay. |

00:23:25 | You do 180 minus 125, you get 55. |

00:23:28 | Well then, if you do 180 minus 55 you get 125 again. |

00:23:32 | Okay. |

00:23:34 | It's longer, but it proves it. |

00:23:35 | Okay. Well, so let's look at the theorem that it wants us to fill in. |

00:23:39 | The measures of an angle formed by two secants or chords that intersect the interior of the circle is what? |

00:23:48 | Blank the blank of the measures. |

00:23:54 | (inaudible) |

00:23:55 | Which one of those methods would fit into that? We have three different ways, which one would fit into there? |

00:24:02 | Half the sum of the two arcs. |

00:24:05 | Okay. So if you take the sum of the two arcs and you divide them in half, I think that's the one that fits best in there. |

00:24:11 | Any of your methods were pretty good. |

00:24:14 | Okay. Michaela had the two that I was thinking of and then Katie had a third one that was really good. |

00:24:21 | The measures end up being half of whatever those two sums are. |

00:24:31 | Because not all the examples are gonna be as specific as these are to follow through with your other- with the third way to do it. |

00:24:40 | Do you understand what I mean? |

00:24:42 | You could probably do it yourself on every example, but that might be time consuming. |

00:24:47 | All right. Let's look at the very last one. |

00:24:50 | You guys are doing really good with these patterns. |

00:24:55 | B D, measure of arc B D is 200 degrees. Now we have the vertex of two secants outside the circle. |

00:25:04 | Okay. We had it on the circle. We had it inside the circle. Now we're dealing with the outside of the circle. |

00:25:09 | A lot of you guys had this question last week when you were talking about that test. |

00:25:14 | You have measure of arc B D. That's at 200 degrees. |

00:25:18 | And then you have A C, the smaller one right before they intersect at 40 degrees. |

00:25:22 | Then you have the measure of angle one. They give you one hundred. Where did they get that? |

00:25:28 | (inaudible) |

00:25:30 | Measure of angle one. Kind of following right along. Melissa? |

00:25:33 | (inaudible) intercepted arc of- for angle of B D has to be half. |

00:25:36 | Good. Intercepted. |

00:25:37 | Good. So it has to be half. Good. |

00:25:40 | Two. Measure of angle two. They say 20 degrees. What do you think? |

00:25:47 | Go ahead. |

00:25:48 | The same reason as for measured angle. |

00:25:50 | Good. You guys are following right along. |

00:25:51 | The same reason. Okay? The other one's 40. This one's half. It's gotta be 20. |

00:25:56 | And they get A V C. |

00:25:57 | Back to this measure angle A V C. |

00:25:59 | If it's outside the circle. What do you think they- how do you think they got that 80 degrees? |

00:26:05 | Ben? |

00:26:08 | You can do- first you gotta do 180 minus one and that gives you the measure of B C D- |

00:26:17 | And then you do that plus adding the two, and then subtract that from 181, and then you have A V C. |

00:26:27 | Okay. So you were going with that whole large triangle- |

00:26:31 | Yeah. |

00:26:32 | Sum. |

00:26:33 | So he said V C B, if you do that minus angle one, which was 100, that leaves you with what? |

00:26:42 | That leaves you with 80. |

00:26:43 | Eighty. |

00:26:45 | Okay? You have 80- you have measure of angle two, which is- |

00:26:48 | Twenty. |

00:26:49 | Twenty. So that gives you 100 so that angle V- A V C has to be- |

00:26:54 | Eighty. |

00:26:55 | Eighty. That works. Right? That works. |

00:26:58 | Or you can do A C is the (inaudible) outside, the angle is outside the circle. |

00:27:08 | Which one- A C what? The arc? |

00:27:10 | A V C. |

00:27:11 | A V C. Angle A V C. |

00:27:12 | It's outside there, but it intercepts A C. The arc A C- |

00:27:17 | Okay. |

00:27:18 | The arc's 40 degrees, and since angle A V C is 80, maybe it's 40 times- times two, (inaudible). |

00:27:26 | Maybe it's 40 times two? Do you have something to back that up? |

00:27:29 | No? |

00:27:31 | It also intercepts arc B D. |

00:27:35 | Okay. |

00:27:37 | But, if you do B D which equals 200 plus A C (inaudible) 240. |

00:27:44 | Mm-hm. |

00:27:46 | At- no, wait. Do 200 minus 40 and then you get 160. |

00:27:51 | Okay. |

00:27:52 | Divide that by two and that equals 80. |

00:27:55 | Okay. 'Cause we're trying to stick with this general pattern of it ending up being- what's it end up being every time? |

00:28:03 | If you've been following this pattern of the angle it turns out to be- |

00:28:07 | Half. |

00:28:08 | Half- you're scaring me- it ends up being half of whatever- whatever pattern you're following. Okay. |

00:28:17 | The last one we added the two arcs together because it was inside the circle. |

00:28:22 | Now we have a different case. |

00:28:23 | It's still gonna be half, but Leah says it might be half of subtracting them because now you're outside the circle. Okay. |

00:28:31 | So if you're- if you have your- what side's it on? |

00:28:34 | If you have your B D and then you're a C arc here, the angle's way over here. Subtracting them might make sense. |

00:28:40 | Okay? |

00:28:41 | I see. |

00:28:42 | Do you see what I mean? Subtracting them might make a little bit more sense. |

00:28:44 | (inaudible) |

00:28:45 | Okay? |

00:28:46 | And now Ben's idea still worked, but it might be difficult for you. |

00:28:52 | You're trying to get something quick and easy that's a gen- you're following this pattern. The triangle sum works. Right? |

00:28:58 | But that might take you a little extra work every time. |

00:29:02 | If you're only given the measure of two arcs that it intercepts, it's gonna be, probably, a lot easier just to subtract them. |

00:29:12 | Boom. Get your answer. |

00:29:14 | So, go for it Raul. Read that theorem for me and fill in the blanks. |

00:29:20 | The measure of an angle formed by two secants that intersect the exterior of a circle is half the sum of the measures of the intercepted arcs. |

00:29:28 | Half of the sum? |

00:29:30 | (inaudible) |

00:29:32 | Half of the- |

00:29:34 | What's it called? |

00:29:40 | The... |

00:29:41 | Difference. |

00:29:42 | Difference. We're subtracting them this time. |

00:29:44 | I thought I said that. |

00:29:45 | You did say that. I was trying to see if Raul would get that. |

00:29:49 | I thought that angle one was 100 degrees. Right? |

00:29:54 | Okay. |

00:29:55 | And then in order to find A B C, you had- oh, yes. |

00:29:57 | That was Ben's way of the triangle- the triangle sum. We were trying to get down to that same general pattern of it being half. |

00:30:06 | Okay. So if you- if you subtract them, if you get the difference, that's gonna be a quicker way to do it. |

00:30:14 | 'Cause you might not always have that triangle working for you there. |

00:30:19 | That's a little- like you're taking- you're doing it- but you're taking an extra step. |

00:30:22 | Mm-hm. |

00:30:23 | You see what I mean? |

00:30:24 | You're going an extra step. Basically they just want to get the general pattern. It's gonna be a lot easier for you if you subtract them. |

00:30:33 | So you do it by half. Divide it in half. |

00:30:37 | All right. I have an example for you. |

00:30:42 | This is... what we just did with a little twist on it. |

00:30:48 | (inaudible) |

00:30:57 | This is my example with a little twist on it. |

00:31:01 | (inaudible) |

00:31:20 | Given the measure of angle A V C is 60 degrees, the measure of arc A C is a 130 degrees, |

00:31:29 | you have to find the measure of arc B D. |

00:32:32 | I like how you wrote that like that. |

00:32:48 | How did you get your answer there? |

00:32:50 | I used a theorem. |

00:32:52 | What did you do? |

00:32:55 | I just used that theorem. |

00:32:57 | How, though? How did you use it? |

00:32:59 | I don't know. I did something. |

00:33:03 | You did something? |

00:33:04 | All right. |

00:33:25 | Okay. See now I like how you're filling in those blanks. |

00:33:28 | Write it out, show me what you did. |

00:33:41 | You want to go write that up there for me? |

00:33:46 | Take your book with you if you need it. |

00:33:56 | (inaudible) |

00:33:59 | Good, Katie. |

00:34:04 | Some people had to draw a picture. That's good. Draw your own picture. Some people like to draw it and write an equation. |

00:34:12 | Either way you do it, you gotta show me what you did. |

00:34:17 | Ms. Lancour, should we solve it? |

00:34:18 | Yep, I want you to solve it. |

00:34:29 | Leah did hers with an equation? |

00:34:37 | (inaudible) |

00:34:42 | She followed the subtraction, 130 minus X divided by two should give you 60 degrees, she says. |

00:34:53 | She moved the two over and got 120. One thirty minus X equals 120. So she filled in her blank with the X. |

00:35:01 | X has to be 10 degrees. One thirty minus 10 degrees is 120 degrees. |

00:35:06 | Good. |

00:35:10 | Anybody else want to tell a different way that you did it? |

00:35:16 | Nobody? |

00:35:18 | You? |

00:35:19 | No? |

00:35:20 | No? |

00:35:22 | You all did it the exact same way as Leah did it? |

00:35:24 | Yep. |

00:35:25 | That is absolutely amazing. That is absolutely amazing. Nobody can- Okay. I'll let it go. |

00:35:33 | Okay. That one was the- that one was the- the nice one. |

00:35:40 | That one was the nice one. |

00:35:41 | (inaudible) |

00:35:42 | I like nice ones. |

00:35:45 | Not me. |

00:35:46 | Now I don't have to show my work. |

00:35:48 | No, you gotta show the work. |

00:35:50 | (inaudible) |

00:35:55 | You have to actually do it. |

00:36:31 | (inaudible) |

00:36:33 | Okay. That one's good. Good start on that one. |

00:36:43 | What are you gonna do first? Which one are you gonna work on first? The R S U? |

00:36:46 | Yes. |

00:36:50 | R S U looks like the vertex is on the circle. |

00:36:52 | Yes. |

00:36:53 | What can you do for that? |

00:36:56 | (inaudible) |

00:36:58 | Sometimes it helps if you draw a picture. It helps me to draw a picture. |

00:37:21 | Raul, what are you doing over here? |

00:37:24 | Well, I don't have any (inaudible). |

00:37:25 | Which angle are you doing first? |

00:37:27 | Well, I don't- I haven't (inaudible) problem, but I'm doing U T before (inaudible) anything else. Makes it simple for me. |

00:37:34 | Okay. Why do you think that one's easier? Why is that? |

00:37:38 | 'Cause then I feel like the question would be R S U. There's R S U. Oh, that would be half of that. (inaudible) that. |

00:37:45 | Oh, I see. You're finding this arc first. |

00:37:47 | Yeah, this one first. |

00:37:48 | Oh! Okay. I like that. Not bad. |

00:38:00 | How'd you get this one, Melissa? |

00:38:03 | That, it intercepts arc R U, so it has to be half. (inaudible). |

00:38:09 | It has to be half. It's the inscribed angle. Okay. |

00:38:15 | It looks like just about everybody got the first one, R S U. Anybody want to tell me what you did? |

00:38:21 | Katie, did you get that one? R S U. What did you do? |

00:38:25 | Since the point's on the circle, I just divided the arc by two. |

00:38:30 | Okay. Because the R S U, the point, is on the circle- |

00:38:34 | Yeah. |

00:38:35 | So it has to be... |

00:38:37 | Half of the arc. |

00:38:39 | Okay. It has to be half of that intercepted arc. Good. |

00:38:55 | Where is (inaudible)? |

00:38:56 | Where are you, Margaret? |

00:38:58 | Right here. |

00:38:59 | R V U. What did you come up with? |

00:39:01 | (inaudible) |

00:39:07 | R V U. Well, think about what we just did in the- in the last example. |

00:39:13 | When it's outside the circle. You know R U, that was given. And you know R S- oh, not R S. S T. |

00:39:24 | (inaudible) |

00:39:25 | So, you have 140 and you have 30 degrees. |

00:39:33 | Okay, you started to subtract. |

00:39:36 | Well- |

00:39:37 | You changing your mind? |

00:39:38 | No. |

00:39:39 | No? Okay. |

00:39:41 | (inaudible) |

00:39:43 | So, what are you gonna do with that? You got 110. |

00:39:48 | Divide it in half. No. Yeah. No. |

00:39:51 | No, yes, no. Maybe? Yes? |

00:39:54 | Maybe. |

00:39:55 | Look back here. What did we just come up with for this one? |

00:40:04 | Think about it. I'll be back to you. |

00:40:12 | U S V. U S V. How did you get that one, Leah? |

00:40:19 | 'Cause R S U is half of 140, so it's 70. And that's a line (inaudible) equals 180. One eighty minus 70 equals 110. |

00:40:31 | Okay. Okay. |

00:40:38 | Did you follow- |

00:40:44 | Hmm. Show- explain that to me again. Tell me what you just did again. |

00:40:50 | This angle, R S U, is 70- |

00:40:53 | Okay. |

00:40:54 | That's half of that. |

00:40:55 | Okay. |

00:40:56 | And this equals 180, so 180 minus 70 is 110. |

00:41:02 | Oh, okay. You were doing U S V. Okay. Okay. At first- at- at first I thought you were doing the triangle. |

00:41:13 | No. |

00:41:14 | The triangle sum. But you ended up doing... 180 degrees. |

00:41:21 | 'Cause the line equals- |

00:41:22 | Minus... |

00:41:23 | Yeah. |

00:41:24 | Okay. Okay. |

00:41:31 | I like how you did that different. I wasn't expecting that. I'm sorry, that's why I'm thinking. I wasn't expecting that one. |

00:41:39 | Okay. Good, good. |

00:41:42 | Okay. We're gonna go on with R V U really quickly. R V U. |

00:41:51 | This one, R V U. Here. That's kind of like the example we just did, and the very last activity we did. Margaret, go for it. |

00:42:01 | You do 140 degrees- |

00:42:02 | Okay. |

00:42:03 | Minus 30 degrees, and you get 110. So, you divide that in half and get 55 degrees. |

00:42:08 | Good. One thirty- 140 minus 30 gives you 110. Divided in half... 55 degrees. |

00:42:13 | (inaudible) |

00:42:14 | Okay, that's the last theorem we just went over. How about U S V? Let's point that one out. |

00:42:23 | U S V. Going here. I actually got a different way from one person that I w- just really wasn't expecting. |

00:42:32 | I had it one way in my mind, and somebody else gave me a whole different way to think about it. |

00:42:36 | Michaela? |

00:42:37 | I used the triangle sum theorem. |

00:42:38 | Okay. |

00:42:39 | And we just did R V U, so I know that that is 55 degrees. |

00:42:45 | Okay. So she knows that this one is 55. Okay. |

00:42:48 | And since S U V- |

00:42:51 | This one. |

00:42:52 | I used the- oh, what's it called? The arc intercept theorem. |

00:42:55 | Right. Okay. |

00:42:56 | To find out that that was 15. |

00:42:58 | Inscribed angle to get this one. Fifteen, 'cause it's half. Okay. So, so far we have 55 and 15. |

00:43:05 | I added them together and I got 70, so I did 180 minus (inaudible). |

00:43:10 | Okay. She did 180 minus 170, got 110 for this angle here. So she ended up with 110 for this by doing triangle sum. |

00:43:21 | Okay. How about- Leah, tell me what you did, 'cause it just- I was on a whole 'nother different- |

00:43:26 | I was on this inscribed angle and getting the triangle sum theorem. It didn't even occur to me. |

00:43:33 | Angle R S U, we already figured that (inaudible) 70. |

00:43:37 | Okay. We did this, RSU, which was... 70. |

00:43:43 | Because if the R equals 140 and then R line- segment R S V- |

00:43:53 | Okay. |

00:43:54 | (inaudible) that line, the- the angle of it equals 180. |

00:43:58 | Okay. It's a straight angle. It has to be 180. |

00:44:00 | Yeah. |

00:44:01 | And so you do 180 minus 70. You get 110. |

00:44:04 | And she got 110. |

00:44:06 | That's the easy way. |

00:44:07 | That's an easy way. Well, we didn't even- it didn't- it didn't even click in my head. I'm- I'm sure it didn't- took- I was surprised Leah got that. |

00:44:15 | And last one. R W S. |

00:44:19 | Real quick. Don't go. R W S. How would you get that one? |

00:44:26 | Ninety-five. |

00:44:27 | She says 95. You say? |

00:44:28 | Ninety-five. |

00:44:29 | Ninety-five. |

00:44:30 | (inaudible) triangle sum again. |

00:44:31 | You did triangle sum? |

00:44:33 | I did triangle sum. |

00:44:34 | I did was to figure out what arc U T equals. It equals 90. You do 100 plus 90 divided by two. |

00:44:42 | Good. Okay. Thank you, guys. |

00:44:48 | Study, study, study, study, study, study. Tomorrow. |

00:44:51 | Tomorrow? |

00:44:52 | Tomorrow. |

00:44:53 | Tomorrow's the what? |

00:44:54 | Final. |