My Math Forum What is the measure of $\angle AOC$?

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 April 8th, 2018, 06:00 PM #1 Senior Member   Joined: Aug 2014 From: India Posts: 343 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, 08:52 PM #2 Senior Member   Joined: Aug 2014 From: India Posts: 343 Thanks: 1 I assume that the center of the inscribed circle fits on the bisectors of the two given angles.
 April 9th, 2018, 04:56 AM #3 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 895 Seriously? The way you have drawn this AOC is a straight line. Is that correct?
 April 9th, 2018, 05:53 AM #4 Senior Member     Joined: Feb 2010 Posts: 701 Thanks: 136 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, 06:49 AM #5 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,791 Thanks: 970 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.
 April 9th, 2018, 07:24 AM #6 Global Moderator   Joined: Dec 2006 Posts: 20,098 Thanks: 1905 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. Thanks from Denis and Ganesh Ujwal
April 9th, 2018, 08:51 AM   #7
Senior Member

Joined: Aug 2014
From: India

Posts: 343
Thanks: 1

Quote:
 Originally Posted by Denis The diagram makes no sense.
Why you always ready to hate my writing?

April 9th, 2018, 09:49 AM   #8
Math Team

Joined: Oct 2011

Posts: 13,791
Thanks: 970

Agree with ~4.76, Skip: my eyeballing faulty this am!

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

 Tags $angle, aoc$, measure

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