My Math Forum  

Go Back   My Math Forum > High School Math Forum > Geometry

Geometry Geometry Math Forum


Reply
 
LinkBack Thread Tools Display Modes
February 10th, 2015, 08:27 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.
coolikedat99 is offline  
 
February 10th, 2015, 08:40 AM   #2
Math Team
 
Joined: Dec 2013
From: Colombia

Posts: 7,306
Thanks: 2443

Math Focus: Mainly analysis and algebra
It could be a trapezium (the congruent sides being non-parallel).
v8archie is offline  
February 10th, 2015, 08:44 AM   #3
Math Team
 
Joined: Dec 2013
From: Colombia

Posts: 7,306
Thanks: 2443

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.
v8archie is offline  
February 10th, 2015, 11:18 AM   #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).
coolikedat99 is offline  
February 10th, 2015, 01:08 PM   #5
Global Moderator
 
Joined: May 2007

Posts: 6,510
Thanks: 584

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.
mathman is offline  
February 10th, 2015, 02:21 PM   #6
Global Moderator
 
Joined: Dec 2006

Posts: 18,956
Thanks: 1603

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.
skipjack is offline  
February 10th, 2015, 02:37 PM   #7
Math Team
 
Joined: Dec 2013
From: Colombia

Posts: 7,306
Thanks: 2443

Math Focus: Mainly analysis and algebra
This something I had in mind. I suspect that the quadrilateral is not convex.
v8archie is offline  
February 10th, 2015, 04:46 PM   #8
Global Moderator
 
Joined: Dec 2006

Posts: 18,956
Thanks: 1603

It can be convex, but the construction I gave doesn't ensure that it's convex.
NonParallogram.PNG
skipjack is offline  
Reply

  My Math Forum > High School Math Forum > Geometry

Tags
counterexample, parallelogram



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Finding Counterexample's With Sets? LaC0saNostra Applied Math 2 September 30th, 2012 02:43 PM
is this proof by counterexample right? Solarmew Applied Math 2 November 3rd, 2011 07:08 PM
What is the counterexample here? Singularity Number Theory 5 February 26th, 2011 09:28 AM
Counterexample Help julian21 Number Theory 1 September 28th, 2010 11:43 PM
Give counterexample in real analysis weier Real Analysis 2 December 4th, 2006 01:43 PM





Copyright © 2018 My Math Forum. All rights reserved.