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Geometry - Arcs created by Secant Lines

  1. Apr 25, 2017 #1
    1. The problem statement, all variables and given/known data

    This picture:

    It's drawn with exactly the amount of information from the worksheet. Specifically, the two secants meet at a point, with an angle of 28 degrees between them. Both secants partition off an arc of 120 degrees. The goal is to find the arc labeled x.

    2. Relevant equations

    Theorem (1): If an angle's vertex lies on a circle, then the angle is equal to half of the subtended arc.
    Theorem (2): An angle is a central angle IFF the subtended arc is equal to the measure of the angle.

    3. The attempt at a solution

    Visual aid:

    I labeled the points where the secants touch the circle A, B, C, D I then drew the purple lines.

    By theorem 1, angles ADB, CBD are both 60 degrees.

    Thus angle BED is 60 degrees since BED is a triangle, formed out of three straight lines.

    Since AD is a straight line, AEB + BED = 180 degrees.

    Thus AEB = 120 degrees.

    This violates Theorem 2 since E is clearly not the center of the circle. I'm honestly not sure what I'm doing wrong. I've stared at this on and off for a couple hours now. Thanks for any directions.
  2. jcsd
  3. Apr 25, 2017 #2


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    Homework Helper

    Use the central angles. Theorem 2 is not true for angle AEB. What is subtended arc?
    Last edited: Apr 25, 2017
  4. Apr 27, 2017 #3


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    Some more hints:

    Determine the green angles, then the red ones. With the 28° angle and the red ones, you get the blue angle, and then x.
  5. Apr 27, 2017 #4
    Thank you. I guess my problem is I forgot that I could manufacture central angles even though the figure didn't contain the center of the circle.
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