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Math Help - Circles and Angles

  1. #1
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    Circles and Angles

    I'm hoping that someone can give me some help with the following problem:

    O is the centre of the circle (show on the attachment to this post) and AB is parallel to CD. Find the angles labelled x and y.

    With a bit of cheating I was able to find that x = 32o and y = 58o but my questions is 'why' or what is the 'proof' of this?

    Many Thanks.
    Attached Thumbnails Attached Thumbnails Circles and Angles-cir-quest.jpg  
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  2. #2
    MHF Contributor ebaines's Avatar
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    Re: Circles and Angles

    If you construct a line from Point B horizontally to intersect DC at right angles at Point E, from symmetry you can see that the length of that line is R sin(64). And from that the angle BDC (which also equals angle x) is arctan(Rsin(64)/(R+Rcos(64)). From that you have angle x.

    Length CE is then R-Rcos(64), so angle y is arctan(Rsin(64)/(R-Rcos(64)).

    Circles and Angles-angles.jpg
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  3. #3
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    Re: Circles and Angles

    Circles and Angles-circle.jpg

    ADO = ODA = 58 (isosceles triangle)
    DBC = Right angle (Angle on a semi circle)
    ABD = 180 -(90 + 58) = 32 (Cyclic quadrilaterals opposite angles are supplementary) [Angle x = 32]
    ABD + DBC +BCD = 180 (Supplementary angles between parallel lines)
    ABD (32) + DBC (90)= 122 therefore BCD = 180 = 122 = 58 [Angle y = 58]
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  4. #4
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    Re: Circles and Angles

    Quote Originally Posted by wardr1 View Post
    I'm hoping that someone can give me some help with the following problem:

    O is the centre of the circle (show on the attachment to this post) and AB is parallel to CD. Find the angles labelled x and y.

    With a bit of cheating I was able to find that x = 32o and y = 58o but my questions is 'why' or what is the 'proof' of this?

    Many Thanks.
    Circles and Angles-untitled2.gif
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