My Math Forum Counterexample for Parallelogram

 Geometry Geometry Math Forum

 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: 7,553 Thanks: 2556 Math Focus: Mainly analysis and algebra It could be a trapezium (the congruent sides being non-parallel).
 February 10th, 2015, 09:44 AM #3 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,553 Thanks: 2556 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: 6,660 Thanks: 648 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: 20,096 Thanks: 1905 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: 7,553 Thanks: 2556 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: 20,096 Thanks: 1905 It can be convex, but the construction I gave doesn't ensure that it's convex. NonParallogram.PNG

,

### parallrlogtams counterexample

Click on a term to search for related topics.
 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post LaC0saNostra Applied Math 2 September 30th, 2012 03:43 PM Solarmew Applied Math 2 November 3rd, 2011 08:08 PM Singularity Number Theory 5 February 26th, 2011 10:28 AM julian21 Number Theory 1 September 29th, 2010 12:43 AM weier Real Analysis 2 December 4th, 2006 02:43 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top