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Math Help - shaded/unshaded regions

  1. #1
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    shaded/unshaded regions

    I'm stuck proving that the answer to this question is 60. I know that a shaded region + unshaded region would have to add up to 180 but I don't see what to do next?
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  2. #2
    Super Member TheChaz's Avatar
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    Let's call the supplementary angle y.
    Then, as you noticed, x + y = 180.

    Since the areas are proportional to the angles, we can just reword the rest in terms of angles. i.e.
    "The sum of the angles of the unshaded regions is equal to the average of the angles of the shaded regions"
    The sum of the unshaded angles is x + x = 2x.
    The average of the shaded angles is (y + y)/2 = y.
    These are equal, so 2x = y.
    Combining this with x + y = 180 will lead you to your desired result.
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  3. #3
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    I thought about doing the same thing Chaz--but how do we know the areas are proportional to their angles?
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  4. #4
    Super Member TheChaz's Avatar
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    Since the vertex is actualy the center of the circle, angles subtend proportional segments, etc.
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