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April 8th, 2018, 05:00 PM   #1
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What is the measure of $\angle AOC$?

A circle with center O is inscribed in a quadrilateral ABCD. AB// CD, $\angle BCD$ = 60 and $\angle ADC$ = 40. What is the measure of $\angle$ AOC ?



I have no idea how to proceed from here. I drew this diagram.

Answer of this problem: 140.
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April 8th, 2018, 07:52 PM   #2
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I assume that the center of the inscribed circle fits on the bisectors of the two given angles.
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April 9th, 2018, 03:56 AM   #3
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Seriously? The way you have drawn this AOC is a straight line. Is that correct?
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April 9th, 2018, 04:53 AM   #4
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Draw a radius from $\displaystyle O$ to $\displaystyle AD$ hitting at point $\displaystyle X$. Draw a radius from $\displaystyle O$ to $\displaystyle DC$ hitting at point $\displaystyle Y$. Then $\displaystyle \angle XOY=140$. No idea what $\displaystyle \angle AOC$ is.
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April 9th, 2018, 05:49 AM   #5
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The diagram makes no sense.
CD should be approximately 6 times longer than AB!
Then, with E and F on CD, draw perpendiculars AE and BF.
Resulting triangles BCF = 30-60-90 and ADE = 40-50-90.
Then easily shown that angleAOC = 140.
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April 9th, 2018, 06:24 AM   #6
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QuadCircle.PNG
As AB$\parallel$CD, $\small\angle$DAB = 180$^\circ$ - 40$^\circ$ = 140$^\circ$.
Similarly, $\small\angle$ABC = 120$^\circ$.
As OA and OC bisect angles DAB and BCD respectively,
$\small\angle$AOC = 360$^\circ$ - 140$^\circ$/2 - 120$^\circ$ - 60$^\circ$/2 = 140$^\circ$.

CD is about 4.76 times as long as AB.
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April 9th, 2018, 07:51 AM   #7
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Quote:
Originally Posted by Denis View Post
The diagram makes no sense.
Why you always ready to hate my writing?
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April 9th, 2018, 08:49 AM   #8
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Agree with ~4.76, Skip: my eyeballing faulty this am!

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Originally Posted by Ganesh Ujwal View Post
Why you always ready to hate my writing?
VERY sorry. I should have said "the diagram is not to scale".
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