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 April 8th, 2018, 05:00 PM #1 Senior Member   Joined: Aug 2014 From: India Posts: 458 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: 458 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: 3,264 Thanks: 902 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: 711 Thanks: 147 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: 14,597 Thanks: 1038 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, 06:24 AM #6 Global Moderator   Joined: Dec 2006 Posts: 20,921 Thanks: 2203 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, 07:51 AM   #7
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Joined: Aug 2014
From: India

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Quote:
 Originally Posted by Denis The diagram makes no sense.
Why you always ready to hate my writing? April 9th, 2018, 08:49 AM   #8
Math Team

Joined: Oct 2011
From: Ottawa Ontario, Canada

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Thanks: 1038

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 Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post EvanJ Calculus 1 July 24th, 2017 06:02 PM putyika Geometry 2 December 5th, 2016 10:32 AM GIjoefan1976 Algebra 4 February 16th, 2016 12:55 PM mared Geometry 6 March 23rd, 2015 05:42 PM kingkos Algebra 18 December 21st, 2012 02:18 AM

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