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Added Apr 05, 2015 •

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10.5 angle relationships in circles now quick review we learned about two different kind of angles before we learned about triangles which The Sinner or the vertex is on the center and the central angle is equal to the arc measure so whatever the angle equals art does and then we learned about inscribed unpause in that was the vertex was on the circle and the sides were made of cords and the angle was equal to half of the arc measure now we're going to use both of those as well as the new stuff now as far as the new stuff goes we've got a couple of new theorems and ways to find some different angles and Arc measures given different situation snow the first one is when you have a tangent and a chord  know if a tangent in a chord intersecting a point on the circle so they intersected a right here then the measure of each angle formed is half the measure of its intercepted Arc so angle 2 is going half of this art and angle one is going to be half of this Arc so the angles equal half of the intercepted Arc so here's an example right here this is 80 no remember the angle is half so the Arts going to be double that so the ark is going to be 160 and this Arc is 200 so angle right here across from it is going to be 100 and also notice this is a straight line it equals 180 Circle past equal 360 200 in 160 equal 360 when two lines intersect in a circle there are three ways make an intersect they can intersect on the circle they can intersect inside the circle or they can intersect outside the circle now on the circle this is just inscribed angle  and we already know how to find that because this angle right here is half of this Arc so this one we already know how to do okay but we need to focus on if it intersects inside and if it intersects outside so we have tooth one for each of these situations now Theron 10.12 is going to be the first one if two chords intersect inside the circle turn the measure of each angle is one half the sum of the measures of the arcs there's your Arc intercepted by the angle vertical angle okay so if you want to find angle one or angle to that angle which they're equal because they're vertical those shuttles are equal to half the sum of the intercepted Arc so if I were to write a literal equation for this right here I would say that angle too is equal to one half  Arc BC right here plus Mark D which is right here and then angle 1 or angle 2  are both equal to that song okay will do an example that in just a second then we have theorem 1013 which is if they intersect on the outside of the circle so if I change it in a secant which would be this situation two tangents which would be this situation or to see what's which would be this situation intersect outside of the circle then the measure of the angles form is one half of the difference of the measures of intercepted Arc so if it's inside it's the sum and if it's outside it's the difference so adding and subtracting is the only difference here so the angle that you're trying to find is equal to half the difference of the intercepted arcs now remember intercepted Arts are the ones on the interior so say right here we wanted to find angle F what we would do would be half of  the arc XY minus the arc GH so it's this Arc - this Arc so that would be the formula to fund this off right here and likewise it would be the same thing right here that you would do this Arc minus this Arc to find that Google and if you'll notice here these have that up so this Arc - of this art will give you that angle all right  okay here's an example of an inside they cross inside so if we need to find these angles right here all you have to do is add these two arcs together and then / 2 basically so your angle is 1/2 of 35 + 80 so that's 1/2 of 115 which is 57.5 degrees that makes each one of these 57.5 degrees 57.5 degrees now just because we know these are vertical we could also I'm these two angles to so if we subtract from 180 then we know that both of these are going to be 122.51 22.5 now there is no formula just because we know these two angles that we get these two Arts we just know what they have to add up to but we don't know what each one used to be  okay now on this one we have outside angle so we need to find this angle out here and we know this Arc is intercepted and this Arc is septic never remember when it's outside you're going to subtract instead of add so we're going to take half of 120 68 which is 52 which is equal to 26 degrees this angle right here 26 degrees and then of course that one's 26 you could subtract from 180 and find those two it's an exterior angle and if you notice it's two tangents that intersect outside now remember in a previous lesson we did learn that these two distances but we want to find the angle so if this angle or this Arc is 250 we subtract them / 2 in find that one but what is this bark have to be well the whole circle has to be 360 so to fund this argue would do 360 minus the 250 and then would get 110 so this one right here has to be 1:10 so you'll do 1/2 it's on the outside so we subtract 2 - 110  which would be half of 140 and 1/2 of 140 is 70 degrees so this right here is 70° alright the next one that's a tangent and secant so on that one that's not the one half of the subtracting or adding remember that this angle is half of the arcade intercepts at this angle is half of the arc in intercept we need there's two different ways you can do this half of one I mean half of 220 is 110 degrees now you know these two are linear so strike for 180 are you know these have to add up to 360 so you could find with this Arceus and take half of that either way that you do it you're going to get 70 degrees for this angle and then you can also find that this art would have to be 140 degrees right so look at the next when it's a little more complicated we got all kinds of stuff going on weave what one's it intersect inside right here in right here we've got one that intersects on the circle right there and then we have an arc  measure right here right here and that involves 40 is what is given can you find all the rest of the angles and arcs in this picture now I want you to think and look real okay the first thing I want you to notice is this is an inscribed angle right here so this angle 40 this Arc is double that because then go his half of the art so right play we know that that one is 80 degrees okay then look at this and you know that this art right here from here to 80 + 30 will come into play to okay so the next thing that you might want to find is these two lost because you know this intercepted Arc and you know this intercepted Arc and they go with these two angles right here and you know that this is inside so it you add the two are 1/2 half of the Sun so when we do that and we add 30 + 76 and then we divided by 2 we're going to get 50 so 53 here 53 here okay then we can subtract from 180 and get 127 here can 127 here right now let's look at the other ones first off there's all kinds of things you could do if you noticed this what's the Little Triangle you could actually use that to find that angle so 53 + 40 subtract from 180 and you will get 80 right there these are vertical so that's 87 and then supplementary subtract from 180 this one's 93 10 this one is 93 degrees now the only thing we can't find is this art and this art and of course we know what they're so have to be lady 7 doubled but we don't know what they each need to be and we don't this angle does make that Arc right there but it's not percepted in it's not a central angle so that doesn't help us so this is everything we can find on there everything but those two Arts is the sinner so that makes a central angle right here so if that's 40 what is this central angle have to be central angles are equal so that is 40 which means this one is 40 you got supplementary so this one is 140 and we know this one is 140 while now we got central angles so if these are all sin triangles they equal the ark that they make so this is 140 This is 40 right here and then spark is 140 on the next thing you can find is this outside angle right here because you have the art that's made by it right here and right here so 40-20 is 20 and 1/2 of 20 is 10 so we know that this little angle right here is 10 degree okay now look at  this is a diameter in a tangent what do you know that a diameter in a tangent do to each other they are perpendicular so we know that this is 90° in this site is 90° the can we find each of these angles of that night will look you have a big triangle right here this angles 10 this is 140 that makes 150 so someone 80 and you get the this angle right here has to be 30 so this one has to be 60 to make 90 the only angle that we don't have labeled yet is this one right here but we have a right triangle again right here the singles 40 that's 90 so the Google has to be 50  and now we have every Ark that one that one that one that one that one like old and all lower balls labeled so we've completed that diagram on this start off by looking since we know that's 45 so we can find all of these angles this one will be 45 and subtract from 119 that's 1:35 that's 1:35 we know that these inside angles is this art plus this art / 2 we know everything but one Ark we can figure stop what it would have to be now algebraically that 45 degrees is equal to one half of  first start second Arc or another way to write that would be 45 equals 60 + x / 2 and if you go ahead and multiply the 45 by 2 then you would get 90 Eccles 60 + x subtract the 60 over here x equals 30 because 30  + 60 / 2 is 45 which gives us these angle ride here so now looks like this angles 30 we know it intercepts this Sark and this art so if we had these two arcs to get 80 + 30 is 110 so we know that 110 - this art / 2 equals 30 if you write that out of your Bradley 110 - we don't know / 2 equals 30 degrees x the to get the 60 10 - x if you subtract the 110 over here you still have the negative x negative x equals negative 50 so x equals 50 so we know that this sang arkwright here has to be 50 degrees can I use the center right here and I want you to find as much as you can you should be able to find several angles and several Arts but what I'm going to do is I want to wait want to do this one in class together but I want you to try it it home by yourself first so see what you can find here and then next time in class we will go over this problem together

Uploaded by Pam Chastain.

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