February 10th, 2015, 09:27 AM  #1 
Newbie Joined: Feb 2015 From: Texas, United States Posts: 2 Thanks: 0  Counterexample for Parallelogram
I just missed a math problem involving a justification of whether or not a quadrilateral must be a parallelogram. The quadrilateral had one pair of opposite sides that were congruent, and had one pair of congruent angles. Would this quadrilateral be a parallelogram? I can't think of any counterexample, and would like it if one of you could provide a counterexample.

February 10th, 2015, 09:40 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,136 Thanks: 2380 Math Focus: Mainly analysis and algebra 
It could be a trapezium (the congruent sides being nonparallel).

February 10th, 2015, 09:44 AM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,136 Thanks: 2380 Math Focus: Mainly analysis and algebra 
I suspect you could make an irregular quadrilateral if the congruent angles were at the two ends of one side.

February 10th, 2015, 12:18 PM  #4 
Newbie Joined: Feb 2015 From: Texas, United States Posts: 2 Thanks: 0  Reply
This would be a reasonable counterexample, except that the congruent angles are also opposite from each other (I forgot to mention this).

February 10th, 2015, 02:08 PM  #5 
Global Moderator Joined: May 2007 Posts: 6,433 Thanks: 561 
Part way to solution. Add diagonal (not splitting the equal angles). The two triangles are "almost" congruent (SSA has two possibilities). Show that the alternate possibility can't hold.

February 10th, 2015, 03:21 PM  #6 
Global Moderator Joined: Dec 2006 Posts: 18,544 Thanks: 1476 
Draw a parallelogram in which the known angle is acute and the added diagonal (that doesn't split the known angles) is shorter than the known side. This will make sure that the alternate possibility referred to above does exist and leads to the counterexample being sought.

February 10th, 2015, 03:37 PM  #7 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,136 Thanks: 2380 Math Focus: Mainly analysis and algebra 
This something I had in mind. I suspect that the quadrilateral is not convex.

February 10th, 2015, 05:46 PM  #8 
Global Moderator Joined: Dec 2006 Posts: 18,544 Thanks: 1476 
It can be convex, but the construction I gave doesn't ensure that it's convex. NonParallogram.PNG 

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