find the measure of angle

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April 5th 2012, 10:27 AM
Mhmh96

finding the measure of an angle

http://im23.gulfup.com/2012-04-05/1333649884331.jpg تحميل صور

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?
April 5th 2012, 11:28 AM
earboth

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Re: finding the measure of an angle

Quote:

Originally Posted by Mhmh96

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?)
April 5th 2012, 11:34 AM
Mhmh96

Re: find the measure of angle

Yes i am sure ,the figure is not precise .
April 5th 2012, 12:00 PM
earboth

Re: find the measure of angle

Quote:

Originally Posted by Mhmh96

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.
April 5th 2012, 09:07 PM
earboth

Re: finding the measure of an angle

Quote:

Originally Posted by Mhmh96

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

$\angle(APE) = 360^\circ - 160^\circ - 2 \cdot 90^\circ = 20^\circ$
April 6th 2012, 01:24 AM
Mhmh96

Re: finding the measure of an angle

Thank you very much,what is "a common axis of symmetry"?
April 6th 2012, 01:52 AM
earboth

1 Attachment(s)

Re: finding the measure of an angle

Quote:

Originally Posted by Mhmh96

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.