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Math Help - Calculate X in a circle

  1. #1
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    Calculate X in a circle

    I'm having some difficulties trying to solve this problem seen in the picture below. I just don't know where to begin. Any help would be very appreciated, thanks.

    Calculate the angel X in the circle

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  2. #2
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    Re: Calculate X in a circle

    Quote Originally Posted by gingerale View Post
    I'm having some difficulties trying to solve this problem seen in the picture below.
    Calculate the angel X in the circle
    .
    Don't you know that x+58=180~?
    Why is that true?
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  3. #3
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    Re: Calculate X in a circle

    So the opposite angel is a reflection that together is equal to 180? X = 122, and the angel that is unmarked would then be 77, all together 360. Makes sense. Why is the circle there then, just for confusing?
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  4. #4
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    Re: Calculate X in a circle

    Hello, gingerale!

    Plato is absolutely correct.


    Calculate the angel X in the circle.


    Code:
                  B 
                  o * *
              *           *
            *               *
           *                 o C
    
        A o                   *
          *         +         *
          *                   *
    
           *                 *
            *               *
            D o           *
                  * * *
    We have cyclic quadrilateral ABCD.
    Draw chords AB, BC, CD, DA.

    \text{We have: }\:\begin{Bmatrix}\angle A &=& \frac{1}{2}\overarc{BCD} \\ \angle C &=& \frac{1}{2}\overarc{DAB} \end{Bmatrix}

    \text{Hence: }\:\angle A + \angle C \;=\;\tfrac{1}{2}\overarc{BCD}  + \tfrac{1}{2}\overarc{DAB} \;=\; \tfrac{1}{2}(\overarc{BCD} + \overarc{DAB}) \;=\; \tfrac{1}{2}(360^o)

    \text{Therefore: }\:\angle A + \angle C \;=\;180^o


    \text{Opposite angles of a cyclic quadrilateral are supplementary.}
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  5. #5
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    Re: Calculate X in a circle

    Generally speaking it is true that the measure of an angle, with vertex on a circle, is half the angle measure of the arc it subtends. Here, the angle opposite x has measure 58 degrees and so subtends an arc of angle 116 degrees. Then entire circle has 360 degrees so that leaves 360- 116= 144 degrees for the arc subtended by angle x.
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