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Math Help - find the measure of angle

  1. #1
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    finding the measure of an angle

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    in the figure above O is the center of the circle

    The segment PA is tangent to the circle

    also:
    AG parallel to HF
    the angle FOH=140 degree
    the angle AOH=30 degree
    DOG=DOF

    what is the measure of angle P?
    Last edited by Mhmh96; April 5th 2012 at 11:41 AM.
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  2. #2
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    Re: finding the measure of an angle

    Quote Originally Posted by Mhmh96 View Post

    in the figure above O is the center of the circle

    The segment PA is tangent to the circle

    also:
    AG parallel to HF
    the angle FOH=140 degree
    the angle AOH=30 degree
    DOG=DOF <--- is this correct? Do you mean perhaps DOG = GOF?

    what is the measure of angle P?
    Are you sure that the given values are correct? With the attached sketch I've used my "correction".

    1. The indicated blue angles are right angles (why?)

    2. \angle(GOA) = 2 \cdot \angle(GAP) (why?)
    Attached Thumbnails Attached Thumbnails find the measure of angle-winkelbeip.png  
    Last edited by earboth; April 5th 2012 at 01:01 PM. Reason: diagram added
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    Re: find the measure of angle

    Yes i am sure ,the figure is not precise .
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    Re: find the measure of angle

    Quote Originally Posted by Mhmh96 View Post
    Yes i am sure ,the figure is not precise .
    Sorry I didn't see your reply. Have a look at my edited post. You'll find a diagram drawn to scale with some hints to evaluate the angle in question.
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  5. #5
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    Re: finding the measure of an angle

    Quote Originally Posted by Mhmh96 View Post

    in the figure above O is the center of the circle

    The segment PA is tangent to the circle

    also:
    AG parallel to HF
    the angle FOH=140 degree
    the angle AOH=30 degree
    DOG=DOF

    what is the measure of angle P?
    1. Since \overline{GA} \parallel \overline{FH} and these line segments are chords of the same circle there exists a common axis of symmetry passing through O.

    2. Therefore \angle(FOG) = \angle(AOH) and consequently the angle

    \angle(GOA) = 360^\circ - 140^\circ - 2 \cdot 30^\circ = 160^\circ

    3. The sum of the interior angles in a quadrilateral is 360. Therefore

    \angle(APE) = 360^\circ - 160^\circ - 2 \cdot 90^\circ = 20^\circ
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    Re: finding the measure of an angle

    Thank you very much,what is "a common axis of symmetry"?
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  7. #7
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    Re: finding the measure of an angle

    Quote Originally Posted by Mhmh96 View Post
    Thank you very much,what is "a common axis of symmetry"?
    The perpendicular bisector of one chord is the perpendicular bisector of the other chord too. This bisector is a symmetry-axis of both chords, of the circle and consequently of the two angles of 30 which are left unmarked.

    The symmetry-axis is drawn in red in the attached diagram.
    Attached Thumbnails Attached Thumbnails find the measure of angle-commonaxis.png  
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