Notes 10.5 More Angles in Circles

[00:00:00]hey guys today we're going to talk about notes 10.5 which is on more angles in the circles so to start us off we're going to property is actually pretty similar to the last lesson so we have the tangent and chord angle so it says if a tangent and a chord intersect on the circle the measure of the angle is half the intercepted arc all right so putting this into words here and putting these words into an equation what i mean by that is that the measure of angle one if you look at it over here it's the red part the measure of angle one is actually equal to half of the arc and that arc is that measure of arc a b so it's half of its arc it's very similar to the inscribed angle but here we have kind of an angle that's like made up of a chord like we have there and then the tangent so that's what we get and it works no matter either way you look at it so i have the red part of it there and also have the blue part of it which is that the measure of angle two is actually equal to half of its respective arc but of course its arc of the other side so at that arc is the arc bca because we have the major arc on there on the outside going the other way and that's the new property for the day so let's put this into practice let's try a few examples so our first one here says to find the measure of angle one and the only thing they gave us was the 122 right there so probably the way to go about this is to go ahead and say well i know a circle the total circle is 360 degrees and so i can take 360 and i can subtract the 122 and i if i do that i'd end up at 238 and that is going to be the remaining arc of this circle so all of this space here all of this is 238 degrees okay well because of how um this is set up now it's set up just like our our um example right above right in the notes and so with that being the case we know that this angle one is half of the respect to arc right so we can say that measure of angle one is equal to half of the arc well again we are looking for the measure of angle one and we know the arc is 238 degrees and so we take half of 238 well then we quickly figure out that the measure of angle one is quite simply 119 degrees and that's all there is to this one all right if you look at example b it works very similar to this that one's for you guys to try in your quick check just do like i did in example a and you'll be all right let's go on to another example except number two you want to find the value of x i want to show you that you can actually do these problems in two different directions here so i have my method one and my method two over here on the right i'm gonna show you the same problem we're getting the same answer different ways okay all right so our first strategy is to kind of use this new property and um use it straight up so what we're going to do is we're going to take this 109 and we're going to find its arc and it's our because this piece right here i'm going all the way around and we know that that's just going to be timesing by 2. so 109 times by 2 gives us that arc is 218 degrees while you know a full circle is spring to 60 x makes up the remainder of the circle so x plus 218 must make 360 degrees which is a full circle if you subtract the 218 from both sides you figure out quickly that x is 142 degrees that's all there is to this one now in a similar manner you may have looked at this problem a little bit differently maybe you saw this straight line right down here and you know that straight angles are 180 degrees and so because you know that you know that the x sorry that x can't use x it's something else there but we can say this unknown just get the question right there that unknown plus 109 makes a straight angle must make 180 and so then i can quickly figure out that angle just by subtracting 109 from both sides and when i do i get 71. so we can scribble this out and write a 71 degrees there and if you want to get to the arc we know to get to the arc from the angle we just times by 2.

[00:03:43]so of course we take 71 by 2 71 times 2 that's where we're getting our x value and 71 times 2 you guessed it 142 just like up above all right as you work example b please recognize that you can use either method here and it will work out for you all right our new um a new property is with two chords so if two chords intersect in the interior of the circle then each angle is half of the sum of the intercepted arcs all right so this one's a little bit more complicated but here's what we have here we have this angle one and if you notice this angle one all right it's in this interior of the circle and we have these two arcs that kind of making up this angle one here i get my green marker out here we have this arc a b and we have this other angle which is pink we have this arc cd if we add those and again we're going to take the half of it but if we add those and take half of it that's what the measure of angle 1 would be and that's where your formula comes from so again just explain this one more time here we have two chords the two lines c b and a d and they're intersecting in the circle they form an angle it's in the interior of the circle that interior angle is equal to half the sum of the two respective arcs which is a b and c d all right let's see this problem let's see this in action let's go on down here to example number three we're gonna find the measure of angle one all right so this is quite literally the problem uh just kind of explained up above so we can go ahead and say let's actually use the same colors from above let's grab right here so the measure of angle one is equal to the two arcs respectively added together so we have the green arc which is 25 plus the other arc which is in pink 101 and we divided it by two all right well all you have to do is a little bit of arithmetic here 25 plus 101 it's of course going to give us to um what 126 there and we want to divide that by 2 126 divided by 2 to give us the measure of angle 1 is of course 63 degrees you'll find example b follows very similar to this please follow my example as a reference let's look at example number four this one says to find the measurement one now you might notice this one's different i've also got a two in there right so one is clearly not the angle made up of the two arcs i have here now there is another way of doing this problem i'm actually not going to do that for you right now i'm going to do what i know and here's what we're going to do we're going to go ahead and take a look at this and say well 2 is kind of the angle that forms these two arcs right so that's the one measure of angle 2 would be equal to those two arcs added together the sum of them divided by two well a little bit a little bit of arithmetic will tell you 68 sorry 48 plus 62 is of course 110 and 110 divided by 2 is 55 degrees so the measure of angle 2 is 55 degrees all right well hopefully you remember back from earlier in geometry when you have a straight line right you know straight lines are 180 degrees so this angle 1 is angle 2 form the straight line or in other words angle 1 plus angle 2 half the sum to 180 but we now know what angle 2 is so angle 1 plus 55 must equal to 180 and as i subtract 55 from both sides i find out the measure of angle one that's 125 degrees example b works very similar to this please use minor the reference move on to the next page here i'm going to look at example number five which says to find the value of x and this one again we're still using that same property we're just seeing all the different possible problems that could occur here all right so let's set it up guys if you notice the angle this time it's actually given to us so our angle right sometimes this form is written out like this the angle is equal to an arc plus an arc all over two and your angle is 62.

[00:07:52]one of your arcs is an x the other arc you have is 75 so x plus 75 divided by 2. now be very careful in solving this problem you got to make sure you're following the correct order of operations here what we want to actually do is get rid of our fraction first i have to divide by 2 so i'm just going to multiply by 2. if i multiply by 2 i'd remove the fraction 2 times 62 getting us to 124.

[00:08:18]on the other side we only have the x plus 75 remaining and all we have to do now is of course subtract 75 from both sides as we subtract 75 we'd find out that x is equal to 49 degrees as you work on example b please understand it is very similar to example a so i want to leave you with one hint just a brief interrupt don't forget what the measure of this angle this symbol is all right what's the measure of this that's your hint all right let's move on to some more properties of circles this property will deal with two secants um so if two secants intersect outside the circle then the measure of the angle formed is half the difference of the intercepted arcs all right so you notice i have a pink arc a green arc and a measure of angle one and then i have the two lines of course secant ac and secant db those are my two secants they're intersecting outside the circle so here's what happens here the measure of angle one would equal two now please make sure you take the bigger minus smaller here uh for this case here so i'm gonna go ahead and take my pink mark which is the measure of angle a b and i'm going to subtract the smaller arc which is the measure of by that cd if i divide that by two that is what the measure of angle one would equal all right so bigger arc minus smaller arc divided by two will give you the angle outside the circle there let's look at this in some problems here i'm going to find the measure of angle one this one is quite literally just the problem from up above from the notes and so here's what you have here you of course have which one i used pink for the big one all right so let's use pink for the big one and green for the small one of course we have our angle one there so our angle one is outside the circle formed by two secants so it says that we take the bigger arc minus the smaller arc we're gonna divided by two as you do 87 minus 25 you're going to go ahead and find the 87 minus 25 is of course going to be 62 over 2 and 62 divided by 2 is 31 degrees so measure of angle 1 is 31 degrees please use mine as an example here as you work example b for your quick check let's look at example number seven says to find the value of x this one again very similar to um to above when we worked a problem we just kind of worked more examples to show you where all your x could really lie inside the problem so take a look here this time i gave you the angle and i didn't give you one of the arcs so as you write out your formula here just be very careful your angle is equal to the bigger arc 171 minus the smaller arc which is x all over 2. and then think about what the order of operations is that you need to do to get this x by itself if you said multiply by 2 you'd be correct as we multiply by 2 we're trying to remove the fraction 2 times 43 gives us 86 and now we're down to just 171 minus x all right now as we go through this we're going to have to do what we have to do here the first thing we have to do is subtract 171. now 86 minus 171 is going to make a negative but just hang in there it'll all work itself out 86 minus 171 will actually take you to negative 85.

[00:11:55]but if you're careful on the other side you notice there's still a negative right there so this is still a negative x and as we go to solve for this x our last step is divide by negative 1.

[00:12:04]as we divide the left side by negative 1 we go back to becoming positive so x is a positive 85 degrees as you break example b please do mind as a reference it'll help you out let's look at our last thing for the day last thing for the day is two tangents or also one secant and one tangent and it says that if two tangents or one tangent one secant intersect outside the circle then the measure of the angle formed is half the difference of the intercepted arcs this is actually very very similar to the one we've just done that's just with different things now you have two tangents or one secant one tangent so again to summarize here measure of angle one the angle outside the circle is equal to the bigger arc which is that abd measure of a b d that arc minus the smaller arc which i have is just a d divided by two half of it all right so let's try this one in a few examples down below i want to find the measure of angle 1 using the property from above so you might notice oh he didn't really give me a number on this one i'm missing this other arc but you know that tool in the back you're pocketed full circle is 360 degrees so you can quickly figure out this part of it that we don't know right now we can figure that out because that unknown plus the 231 to 31 sorry is 360. so we figure out that unknown just by subtracting 231 from both sides as we do we get 129 so we can put 129 degrees in that spot right there now as we want to move in to find the example one or sorry angle one we you do recognize that it's formed by two tangents it is on the outside of the circle so the measure of that angle is equal to the difference of the intercepted arcs so it's 231 minus 129 always do the bigger mass smaller half of it or divided by two now as you go through and do your arithmetic 231 minus 129 will get you down to 102 over two and solving this one last step here one or two divided by two is of course 51 degrees the measure of angle 1 is 51 degrees all right you'll find example b to be very similar to the one i've just worked please use mine as a guide let's look at our last example for the day i want to find the value of x and you'll notice this problem also looks very similar to one above we just have x and another new spot so with this one we have again this is a tangent and a secant and there's the tangent here's the secant and the intercept outside the circle so that angle is equal to the difference of the intercepted arcs all right so the angle is equal to now you got to make a judgment call here which angle is larger or so which arc is larger and hopefully we can all determine that x is the larger arc so do x minus 62 all over two as we solve this using our arithmetic i'm going to go ahead and multiply by two always get rid of that fraction if you can 2 times 40 of course gives you 80. now we have our x minus 62 on the other side as you move in to solve for x the only last step you need to do is just add 62 to both sides and 80 plus 62 makes 142 and that's what x is x is 142 degrees you'll find example b is found in your quick check so make sure you do that one if you guys have any questions don't be afraid to ask us