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Math Help - Angles in cyclic quadrilateral in semi-circle

  1. #1
    Senior Member I-Think's Avatar
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    Angles in cyclic quadrilateral in semi-circle

    Given a cyclic quadrilateral ABCD in a semi-circle, (let AD be the diameter),let AB=BC=1. Draw the line DB.
    Prove that angle ADB= angle BDC

    This was taken as obvious in a solution I read, but it is not at all obvious to me. Help in understanding this result would be appreciated.
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  2. #2
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    Re: Angles in cyclic quadrilateral in semi-circle

    Hello, I-Think!

    \text{Given a cyclic quadrilateral }ABCD\text{ in a semicircle with diamter AD.}
    \text{Let }AB = BC = 1.\;\text{ Draw the line segment }DB.
    \text{Prove that: }\:\angle ADB \,=\, \angle BDC
    Didn't they explain why it is "obvious"?


    Angles ADB and BDC are inscribed angles.
    . . They are measured by one-half their intercepted arcs.

    We are told that chords AB and BC are equal.
    . . Hence, their respective arcs are equal: . \overline{AB} \:=\:\overline{BC}

    Therefore: . \angle ADB \,=\,\angle BDC

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