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Circle Problem: Solve an angle

  1. Jun 8, 2017 #1
    1. The problem statement, all variables and given/known data[​IMG]

    imgur.com/a/bsAKl

    [I didn't know how else to upload an image from my iPad]

    In this problem:
    -Center of the circle=Point F
    -Arc AB= 110 degrees
    -Arc CD= 40 degrees

    Find the measure of angle E.

    2. Relevant equations

    I know of some equations, but I don't know if they apply to this particular problem.

    3. The attempt at a solution

    I tried for about 10 minutes, but I didn't know where to begin. I was absent on the day this was taught (I think) and I don't feel like searching for this in the eBook.

    Anyways, so far I know:
    -Arc AB + Arc CD + something = 360 degrees

    Other than that, I really don't know what else I can do with this info.

    *I know the answer to this is 40 degrees because it was posted later, but I don't know how that is the answer.
    My friend said that whenever he sees a triangle, he does the Triangle Angle Sum Theorem. He said he did 180 degrees - (110 degrees + 40 degrees) to get 40 degrees.
    *I know that's incorrect because:
    (A) That "shape" is not a triangle because AB and CD are arcs, not straight lines.
    (B) The measurements are for the arcs, not the angles of the supposed "triangle". Arc =/= Angle.

    BUT he still got it correct. A bunch of other people did that wrong method but got the correct answer as well, while I left it blank. Please help guys because this is gonna be on the state final as well.
     
  2. jcsd
  3. Jun 8, 2017 #2

    ehild

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    Why is the minus in front of Arc AB? And how much is that "something" then? what does it represent?

    Well, it is utterly incorrect. The method is wrong, as you said. And 180-(110+40) is not 40!
    Look at the picture. What are the angles x?
    upload_2017-6-8_9-10-48.png



    In the triangles FAC and FBC, what are the angles y?
    The line FE halves both the 40 angle and the angle at E. From the green triangle, what do you get for z?
     
  4. Jun 10, 2017 #3
    Sorry for late response. I didn't mean negative arc, I didn't mean to use "-" as a minus sign, but rather a bullet point. Sorry for the confusion. "Something" is the measures of the other two arcs whose measures are not given. (Arc AC and Arc BD).

    Also, I just remembered. In the original problem, Arc AB is a 100 degrees.

    Okay, now for the solution, assuming arc AB is 100 degrees.

    Now I am a little confused with your labelling. Why are angles AFC and BFD? Why are angles FAC and angles FBD equal?
     
  5. Jun 10, 2017 #4

    ehild

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    You are right, they should not be equal unless it is said so. What was written in the text of the problem?
    The figure looked symmetric.
     
    Last edited: Jun 10, 2017
  6. Jun 10, 2017 #5

    ehild

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    If the figure is not symmetric, it can look like the following picture:
    upload_2017-6-10_18-5-37.png

    You can determine the angles of the green and gray triangles, and also the sum x+y. With those, you get z, the angle in question.
     
  7. Jun 10, 2017 #6
    Hmm... I don't remember the original problem because I don't have a copy of it. I jotted down the picture from my memory. Anyways, I looked up this, and I found this on a website. It was hard for me to understand off of it, but I think that is what I am looking for.
    http://www.mathopenref.com/secantangles.html

    It's gives the theorem, but I wanna know how to solve it without plugging it in the formula.
     
  8. Jun 10, 2017 #7

    ehild

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    There are a few triangles on the picture in my previous post. You do not need anything else, but "the sum of angles of any triangle is 180°."
    What are the central angles in the white triangles in terms of x and y? Note that the triangles are isosceles. So what is x+y?
    What are the angles at A and B in the green triangle?
    What are the angles at A and B in the big triangle ABE? So what is the angle at E?
    upload_2017-6-11_2-47-26.png
     
  9. Jun 11, 2017 #8
    Oh. I see it now. But before I continue to solve it, why are the triangles isosceles?
     
  10. Jun 11, 2017 #9

    ehild

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    OA, OB, OC, OD are radii of the circle.
    upload_2017-6-11_10-18-39.png
     
  11. Jun 11, 2017 #10
    OMG, how could I overlook that. I feel stupid now. But thanks, I will attempt to solve it now, now that I know the 4 triangles are isosceles.
    Okay...

    Green Triangle
    Base angles= [(180 degrees -100 degrees)/2] = 40 degrees
    Grey Triangle
    Base angles= [(180 degrees - 40 degrees)/2] = 70 degrees

    So now, I don't know how to figure out the central angles of triangles AOC and BOD. How do I figure those out?
     
  12. Jun 11, 2017 #11

    ehild

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    You can not. But you know the sum of those angles. And write out the relation between the central angles and the base angles x or y, and then you can figure out the sum x+y.
     
  13. Jun 11, 2017 #12
    Sorry, I do not get how to write out the relationship between the central angles of the white triangles and x and y.

    This might be wrong, but I can only figure out these relations.

    180 degrees= [Angle BOD] + 2y
    180 degrees= [Angle AOC] + 2x
    360 degrees= 140 degrees + [Angle AOC] + [Angle BOD]
    220 degrees= [Angle AOC] + [Angle BOD]
    360 degrees= 220 degrees + 2y + 2x

    Is there any critical reasoning/info I am overlooking/missing?
     
  14. Jun 11, 2017 #13

    ehild

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    It is all right. So what is x+y?

    Now look at the triangle ABE.
    upload_2017-6-12_5-14-53.png

    The angle BAE ix 40°+x. The angle ABE is 40°+y. Write up the sum of the angles of triangle ABE, Substitute the value of x+y. What do you get for z?
     
  15. Jun 12, 2017 #14
    Okay. So:

    180 degrees= (40 degrees + x) + (40 degrees + y) + (Z)
    180 degrees= 80 degrees + (x+y) +z

    How do I figure out the value of "x+y"? That's where I am having trouble.
     
  16. Jun 12, 2017 #15

    ehild

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    You wrote in Post#12 that "360 degrees= 220 degrees + 2y + 2x". What is 2x+2y? what is x+y then?
     
  17. Jun 13, 2017 #16
    You mean what it is? When I try to solve it through substitution method?
     
  18. Jun 13, 2017 #17

    ehild

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    I asked what is the numerical value of (2x+2y) if 360 =220 + 2x+2y.
    Think: You add a number to 220 and you get 360. What number did you add?
     
  19. Jun 16, 2017 #18
    Yes I see what you mean, let me explain.
    I know that:
    (2x+2y)=140 degrees

    However, I tried to solve it fully a couple days ago, and I got infinite many solutions. But for your answer, 140 degrees.

    EDIT:
    I just thought of something. Could I simply:
    2x+2y=140 degrees
    To
    1x+1y=70 degrees

    By dividing by two? Is that what you meant?
     
  20. Jun 16, 2017 #19

    ehild

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    Yes, it is. 2x+2y can be written as 2(x+y), by factoring out 2.
    So 2(x+y)=140
    dividing the equation by 2 : x+y=70°
     
  21. Jun 17, 2017 #20
    Nice! So now, back to the main problem.
    180 degrees= z + x + (40 degrees) + (40 degrees) + y = (80 degrees) + (x+y) + z = 150 degrees + z

    Solve:
    180 degrees = 150 degrees + z
    z = 30 degrees

    z = Angle E

    Angle E = 30 degrees

    Is everything correct?

    *If it is, then I guess the problem was 110 degrees for arc AB instead of 100 degrees, which would produce Angle E as 40 degrees, which was the answer.
     
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