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April 8th, 2018, 05:00 PM  #1 
Senior Member Joined: Aug 2014 From: India Posts: 238 Thanks: 1  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. 
April 8th, 2018, 07:52 PM  #2 
Senior Member Joined: Aug 2014 From: India Posts: 238 Thanks: 1 
I assume that the center of the inscribed circle fits on the bisectors of the two given angles.

April 9th, 2018, 03:56 AM  #3 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,165 Thanks: 867 
Seriously? The way you have drawn this AOC is a straight line. Is that correct?

April 9th, 2018, 04:53 AM  #4 
Senior Member Joined: Feb 2010 Posts: 674 Thanks: 127 
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.

April 9th, 2018, 05:49 AM  #5 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 12,421 Thanks: 832 
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 = 306090 and ADE = 405090. Then easily shown that angleAOC = 140. 
April 9th, 2018, 06:24 AM  #6 
Global Moderator Joined: Dec 2006 Posts: 18,962 Thanks: 1606  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. 
April 9th, 2018, 07:51 AM  #7 
Senior Member Joined: Aug 2014 From: India Posts: 238 Thanks: 1  
April 9th, 2018, 08:49 AM  #8 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 12,421 Thanks: 832  

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