March 30th, 2014, 07:20 PM  #21 
Global Moderator Joined: Dec 2006 Posts: 20,746 Thanks: 2133 
It's a limiting case rather than a special case... did someone say otherwise?

March 30th, 2014, 09:13 PM  #22 
Senior Member Joined: Nov 2013 Posts: 247 Thanks: 2 
No. They said that the parallelogram is a special case of a trapezoid because in the quadrilateral family tree it is a branch off of the trapezoid. In the same way the 3 sides equal trapezoid is a special case of the isosceles trapezoid but not only because it branches off of the isosceles trapezoid but also that the word isosceles means at least 2 sides are equal. Imagine a whole world of quadrilaterals and you trying to trace back the genealogy of the quadrilateral. You start off with the youngest one being the square. He says "Well I am bicentric, a rhombus, a rectangle, and a 3 sides equal trapezoid" So you know that those are his parents. You now ask each one of them. Bicentric says "I am tangential and cyclic" Rhombus says "I am a kite and a parallelogram" Rectangle says "I am a parallelogram, a right trapezoid, and an isosceles trapezoid" 3 Sides Equal Trapezoid says "I am an isoscoles trapezoid" So you know that those are the square's grandparents. You now ask each one of them. Tangential says "I am convex" Cyclic says "I am convex" Kite says "I am tangential" Parallelogram says "I am a trapezoid" Right trapezoid says "I am a trapezoid" Isosceles trapezoid says "I am a trapezoid" So you know that those are the square's great grandparents. You now ask each one of them. Convex says "I am a quadrilateral" Trapezoid says "I am convex" Now here is where there is a problem with going back from the square. How can convex be both one of his great grandparents and one of his great great grandparents? You know that the quadrilateral itself is the square's great great grandparent and that he also has as children the complex quadrilateral and the simple which are cousins of varying degrees for each convex quadrilateral. The simple has as his child both the convex and the concave. Last edited by caters; March 30th, 2014 at 09:30 PM. 
March 31st, 2014, 02:46 AM  #23 
Global Moderator Joined: Dec 2006 Posts: 20,746 Thanks: 2133 
Who are "they"? Do "they" define the terms, such as "trapezoid", that they use?

March 31st, 2014, 09:12 AM  #24 
Senior Member Joined: Nov 2013 Posts: 247 Thanks: 2 
They are people who have taught me math such as Sal Khan. They define a trapezoid the same way anybody else does. That is 2 parallel sides and 2 nonparallel sides. They also say that a parallelogram is a type of trapezoid. The parallelogram is a trapezoid yet the definition of trapezoid is 2 parallel sides and 2 nonparallel sides. Here we have just done a proof of contradiction. What we proved is that trapezoids Can have 2 pairs of parallel sides just by knowing that a parallelogram is a trapezoid. 
March 31st, 2014, 09:43 AM  #25 
Global Moderator Joined: Dec 2006 Posts: 20,746 Thanks: 2133 
Using that definition (which I think is standard), a trapezoid must have two nonparallel sides, so a parallelogram is not a type of trapezoid (but it is a limiting case). Your teachers slipped up if they used the wording "type of".

March 31st, 2014, 09:55 AM  #26 
Senior Member Joined: Nov 2013 Posts: 247 Thanks: 2  If you look at a heirarchy of quadrilaterals you will see parallelogram as a branch off of the trapezoid. That means that it is indeed a type of trapezoid just like the right trapezoid and isosceles trapezoid which are also branches off of the trapezoid.

March 31st, 2014, 10:14 AM  #27 
Global Moderator Joined: Dec 2006 Posts: 20,746 Thanks: 2133 
Not for a strict hierarchy, unless limiting cases are allowed (as already explained).

March 31st, 2014, 10:16 AM  #28 
Senior Member Joined: Nov 2013 Posts: 247 Thanks: 2  File:Quadrilateral hierarchy.png  Wikipedia, the free encyclopedia This shows what I just said. That is a parallelogram as a branch off of the trapezoid. 
March 31st, 2014, 10:22 AM  #29 
Global Moderator Joined: Dec 2006 Posts: 20,746 Thanks: 2133 
It's not a branch off in that way unless you allow limiting cases, which is essentially equivalent to changing the definition of trapezoid.

March 31st, 2014, 10:26 AM  #30  
Senior Member Joined: Nov 2013 Posts: 247 Thanks: 2  Quote:
This shows what I just said. That is a parallelogram as a branch off of the trapezoid. And it says below the image this: Taxonomic hierarchy of quadrilaterals. Lower forms are special cases of the higher forms they are connected to. By that definition a parallelogram since in this hierarchy it is connected to the trapezoid it is a special case of the trapezoid.  

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