in this lesson we're going to focus on circles and angles the first type of angle that you need to be familiar with is known as the central angle the central angle with reference to a circle has the vertex on the center of the circle so let's talk about it so let's say this is circle c and let's call this point a b and this is point c so let's say that angle acb is equal to 50 degrees what is the measure of arc a b now acb is a central angle as you can see the vertex is at the center of the circle and that's equal to 50. now the measure of the intercepted arc and the central angle are the same so arc a b is the same as the central angle they're both equal to 50 degrees so that's the first type of angle that you need to be familiar with now the next type of angle that we need to talk about is the inscribed angle so what do you think this angle represents what is the inscribed angle so in this case the vertex is not at the center but actually on a circle so let's say this is a b and c so angle abc is an inscribed angle it's composed of two chords and that is chord a b and chord bc a chord is simply a line segment that connects two end points on a circle now if such a chord passes through the center of a circle then it's known as the diameter now let's say if angle abc is equal to 30 degrees what is the measure of arc ac now you need to know that the inscribed angle is half of the measure of the arc so the arc is twice the value of the inscribed angle so arc ac is 60 degrees so make sure you understand that next let's talk about the tangent chord angle so based on a name this type of angle is formed when a tangent segment meets a chord so let's draw a circle so first let's draw a tangent segment keep in mind a tangent line touches the circle at one point and then let's draw a chord so let's call this a b and c so a b is the chord and bc is a tangent segment now let's say if angle abc is 25 degrees what is the measure of the intercepted arc that is arc a b so this angle is 25 degrees the arc which goes from a to b so that arc is 50 degrees the intercepted arc is twice the value of a tangent chord angle and so that's another rule that you want to write down next up is the chord chord angle so feel free to pause the video and draw a picture that represents the chord chord angle in a circle let's see if you can come up with it so basically this angle is formed from the intersection of two chords so let's call this a b c d and e so notice that angle abc and dbe they're vertical angles so they're congruent now the measure of angle abc is basically the average of arc ac and rd it's one half the sum of arc ac and arc i'm running out of space d e so let's say if the measure of arc ac is 100 degrees and the measure of arc d e is 60 degrees what do you think the measure of angle abc will be well the midpoint of 60 and 100 the middle number is 80. so these two will be 80. and to mathematically show that the measure of arc abc is going to be one half the measure of arc ac which is 100 plus the measure of our de which is 60. now 100 plus 60 is 160 and 160 divided by 2 is 80. so that's the measure of angle abc which is the same as its vertical angle and so that's how you can calculate the chord chord angle if you're given the intercepted arc of the chord chord angle and the arc that comes from the vertical angle which is uh this one now let's work on some problems dealing with the chord chord angles so once again let's say that this is a b c and that's d and e and let's say the measure of arc d e is 70 degrees and angle dbe is 55 what is the measure of arc ac so feel free to pause the video and try it so let's call this x now the chord chord angle the 55 degree angle that's going to be one half of the sum of the two intercepted arcs so it's one half of 70 plus x so we just got to calculate the value of x the first thing i'm going to do is get rid of the fraction so a half times two is one so i no longer need the parentheses on the right side on the left it's 2 times 55 which is 110. now i just need to subtract both sides by 70. and so 110 minus 70 is 40. so the measure of arc ac is 40 degrees so that's the answer let's work on another similar problem so let's draw two intersecting chords let's call this a b c and this is going to be d e and let's say that the measure of arc d e is 110 and the measure of arc ac let's say it's 50. i know it's not drawn to scale but we'll make it work what is the measure of angle ebc so go ahead and try that problem so let's call that angle x and let's call this angle the chord chord angle relative to these two arcs let's say that's y we know that the chord chord angle is the average of the two arc angles that it's associated with so it's going to be one half of 110 plus 50. so 110 plus 50 is 160 and half of 160 is 80. so now we have the value of angle y so that's 80. and notice that these two angles form a linear pair so y plus x is 180. so x is 180 minus y or 180 minus 80. so angle x is 100 degrees therefore the measure of angle ebc is 100 degrees let's work on another problem so once again we're going to have two intersecting chords let's call this a b c d and e and let's say also we have another point this is x and that's the center of the circle so let's say that dc is a diameter on circle x and you're also given that angle abd is equal to 115 degrees and the measure of arc ac is equal to 75 degrees what is the measure of arc ce go ahead and try this problem so angle abd is 115 degrees now notice that these two angles form a linear pair so angle abc has to be 180 minus 115 so 180 minus 115 that's 65 degrees now angle abc and dbe they're vertical angles so they're congruent therefore angle dbe must also be 65 degrees and these two are vertical angles so angle cbe is 115 degrees now we're given that the arc ac is 75 degrees so what is the value of rd so we know that the quarter chord angle which we could say angle abc which is 65 degrees that has to be one half of the two arcs so arc ac is 75 plus the measure of arc de so if we multiply both sides by 2 we can get rid of the fraction a half times 2 is 1 and 2 times 65 is 130. so 130 is equal to 75 plus the measure of arc de so 130 minus 75 is 55. so that's the measure of rde all right let's get rid of this stuff and so this is 55 degrees now we know that dc is the diameter and the arc of a diameter is basically a semi-circle which represents 180 degrees so if arc c d is 180 degrees what do you think the measure of arc a d is well we can write an equation the measure of arc cd is the sum of the measure of arc ac plus the measure of arc a d so ac is 75 cd is 180 so now we can calculate the measure of arc ad it's 180 minus 75 which is 105. so notice that we have the value of three arcs so we can calculate the fourth one a full circle is 360 degrees so to calculate the missing arc arc ce all of the arcs have to add up to 360. so first let's write this equation so let's call this y this is what we're looking for arc ac is 75 a d is 105 d e is 55. and this is 360. so it's going to be 360 minus 55 minus 105 minus 75. so the measure of arc ce is 125 degrees this is the answer now the next three angles that we're going to go over have similar equations so let's start with the secant secant angle so what exactly is a secant a secant line passes through the circle at two points but we're going to draw a secant segment now the two secant segments will have a common endpoint let's call this a b c so the common endpoint is point b and this is going to be d and e so what is the equation that we need to know for this type of problem you need to know that angle b is one half the difference of the intercept of the arcs so it's one half the difference of the measure of arc ac and the measure of arc d e so let's say if arc ac is 110 degrees and de is 60. what is the measure of angle b so it's going to be one half of 110 minus 60. so 110 minus 60 is 50 and half of 50 is 25. so the measure of angle b is 25 in this example now let's move on to our next topic and that is the secant tangent angle so let's draw this thing so first let's start with a circle and let's redo that let's call this a b c and this is going to be d so notice that a b is a secant segment a secant line touches the circle at two points and notice that bc is the tangent segment a tangent line or even a tangent segment touches the circle only at one point so now to calculate angle b it's going to be one half of the difference of the two arcs so that is arc ac and arc dc so let's say that arc ac is 130 degrees and let's say that angle b is 20 actually let's make it 30 degrees what is the measure of arc dc go ahead and try this problem so using this formula angle b is 30 degrees that's equal to one half the measure of arc ac which is 130 minus the measure of arc dc which we're going to call x now let's multiply both sides by two so half times two is one two times thirty is sixty and that's equal to one thirty minus x so i'm going to take this move it to the other side where it's going to change from negative x to positive x and i'm going to move the 60 to the other side so where it's going to become negative 60. so x is 130 minus 60 which is 70 degrees and so that's the measure of arc dc now what about the measure of arc a d now there's only three arcs in this problem and the measure of all the arcs around the circle has to be 360 degrees as we saw in the previous problem so we can say that the measure of arc a d plus the measure of arc ac plus the measure of arc cd all the three arcs has to add up to 360. so let's say the measure of arc 80 is y ac is 130 cd is 70. 130 plus 70 is 200 and 360 minus 200 is 160. so y which represents the measure of arc a d that's equal to 160 degrees and so that's it for this video so arc dc is not 70 degrees rather i was going to say 90 but it's 70 and rkd is 160. now there's one more angle that we need to talk about and it's the tangent tangent angle so draw a picture this angle so keep in mind a tangent segment touches the circle at only one point so let's call this point a and this is going to be point b we're going to have point x somewhere over here and let's say uh actually i want this to be point b and this is going to be point c and then x as well so for this type of situation the measure of angle b is one half the difference of the measure of arc axc that's the major arc minus the measure of the minor arc ac so let's go over an example let's say if the measure of the major arc is 220 degrees what is the measure of angle b so feel free to pause the video and try this problem so this is 220. so notice that the sum of arc axc the major arc plus the minor arc those two arcs make a full circle so those two angles have to add up to 360. so the measure of arc ac is simply 360 minus the measure of the major arc which is 220. so 360 minus 220 is 140 so now that we have that we can calculate the measure of angle b so it's going to be one half the measure of the major arc which is 220 minus the measure of the minor arc which is 140. 220 minus 140 is 80 and half of 80 is 40. so that's the measure of angle b that's the answer now let's work on some problems that is basically a review of the stuff that we learned so let's call this a b c and d so we have circle d so d is the sun of the circle and let's say that angle bdc is 40 degrees so what is the measure of angle a so feel free to try the problem so let's take this one step at a time so let's focus on the central angle bdc so that's 40 degrees if you recall the measure of the intercepted arc and the central angles the same so they're both 40 degrees now in the second part we have an inscribed angle with the vertex on a circle that's angle bac and it has the same intercepted arc arc bc which is 40. now if you recall the inscribed angle is half of the intercepted arc so half of 40 is 20. and so that's the measure of angle a it's 20 degrees now let's consider another problem so let's say this is a b and c and we have sensor d so given circle d and let's say that the measure of arc bc is 60 degrees what is the measure of angle c so what do you think we need to do in this problem so arc bc is 60 degrees what is the value of arc or rather angle a notice that angle a is the inscribed angle with arc bc being an intercepted arc so the inscribed angle is half of the intercepted arc half of 60 is 30. now d is the diameter i mean rather d is the center of the circle which means that ac is the diameter and so a diameter forms a semicircle with an arc measure of 180. so notice that b is the inscribed angle for arc ac so b has to be half of that which means b is 90. half of 180 is 90. so anytime you have a triangle formed across a diameter this will always be a right triangle it's always going to be 90 because it's always going to be half of 180. so now we can calculate the missing angle angle c so we know that the three angles of a triangle must add up to 180. so 180 minus 90 minus 30. 180 minus 90 is 90 and 90 minus 30 is 60. so the measure of angle c is 60 degrees arc ac is 180 and a b has to be twice this value so 60 times 2 is 120 and we can see that all three arcs which goes around a circle adds up to 360. 120 plus 180 is 300 and then plus 60 that's 360. so you know the work is correct but the answer for the problem is 60 degrees that's the measure of angle c which is what we're looking for let's say this is a b c d and e and let's say that angle a is we'll call it x and angle d is 60 degrees what is the value of x go ahead and pause the video and try this so let's focus on this part so this is a b and e and so notice that angle bae is an inscribed angle which means that the measure of the intercepted arc arc be has to be 2x now let's focus on angle d so we have this picture and so this is b d e so angle d the inscribed angle is 60 degrees which means arc be has to be 60 times 2 or 120. so therefore these two arcs which represent our be they have to be equal to each other so 120 is equal to 2x and if we divide by 2 we can see that x is 60. so in this problem because angle a and angle d they share the same arc these two angles must be congruent to each other and so that's it for this problem let's say this is a b c d and e so arc ac we're going to say that it's 9 x plus 18 and arc d e is 5 x plus 10. and angle abc we're going to say it's x squared plus 6. with this information calculate the measure of arc ac so we know that the measure of angle abc has to be one half the sum of the measure of arc ac plus the measure of arc d e since we have a chord chord angle so angle abc that represents x squared plus six and arc ac is nine x plus eighteen and arg d e is five 5x plus 10. so now what we have is an algebra problem so let's go ahead and solve it so first i'm going to multiply both sides by 2. actually i will need to do that in this example let's combine like terms 9x plus 5x is 14x and then 18 plus 10 is 28. so half of 14 is 7 half of 28 is 14. so this is what we now have now let's take everything from the right side move it to the left so this is going to be x squared minus seven x plus six minus fourteen that's equal to zero now six minus fourteen is negative eight so what two numbers multiply to negative eight but add to negative seven this is going to be negative eight and positive one so we can factor it by writing it this way x minus eight times x plus one and so that's equal to zero so now we could set each factor equal to zero so we have two possible answers for x x can be eight or it can be negative one now it turns out that both answers can work if we choose negative one the arcs will be very small for example arc de would be five times negative one plus ten so it's going to be five now keep in mind both answers are possible but for the sake of simplicity i'm just going to focus on x equals eight so that's just one of the two possible answers you can repeat the steps with the other value of x if you want to so let's calculate the measure of arc ac that's going to be 9 times 8 plus 18. 9 times 8 is 72 72 plus 18 is 90. so 90 is the measure of arc ac now if we want to calculate the other ones we could do so if we want to angle abc is going to be x squared plus 8 i mean x squared plus 6 which is a squared plus 6. so that's 64 plus 6 and so that's 70 degrees and then if we want to calculate arc de that's 5 times 8 plus 10 which is 40 plus 10 so that's 50 degrees now if you want to find the other answer for arc ac just take this value plug it into 9x plus 18. so that's going to be negative 9 plus 18 which is 9 degrees so those are the two possible answers for arc ac it's 90 degrees and 9 degrees let's try one more problem so let's call this a b c d and e so that's a secant secant angle we're going to draw a chord chord angle as well and so let's call this point point f so let's say that the measure of arc ae is 130 degrees and the measure of arc bd is 70 degrees calculate the angle of afe and also uh angle c calculate the measure of these two angles so first let's start with angle a f e a chord chord angle and let's call that angle x i mean or just x and angle c we're going to call it y so x is one half the sum of these two that's for the chord chord angle so 130 plus 70. 130 plus 70 is 200 and half of 200 is 100. so x is 100 degrees which is the measure of angle afe now let's calculate y y is going to be one half the difference of 130 minus 70. so 130 minus 70 is 60 and half of 60 is 30. so 30 is the measure of arc or angle c rather so this is 30. and so that's it for this problem you