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Definitions of Polygons

  1. Aug 13, 2011 #1

    Dembadon

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    Gold Member

    I'm working through the following book: Principles of Mathematics, by Allendoerfer & Oakley.

    Since I haven't taken a proof-based course yet, and won't be able to until spring 2012 :grumpy:, I want to make sure that I'm not forming habits that will hurt me when I do. There are some answers that aren't provided in the back of the book, so I want to check them with you all.

    On Page 5, Problem 2:

    1. The problem statement, all variables and given/known data

    Assume that polygon, side of a polygon, angle, length of side, equal, and parallel, have been previously defined. Then define:

    a) Parallelogram.
    b) Rhombus.
    c) Pentagon.
    d) Regular Pentagon.
    e) Trapezoid.
    f) Hexagon.

    2. Relevant equations

    I believe the exercise wants to ensure that I'm using the "if and only if" bi-conditional logical connective correctly. My understanding is that the "if" includes all of the cases that follow the next clause, and "only if" excludes all others. So, I need to make sure that my definitions do not include other polygons.

    3. The attempt at a solution

    Parallelogram: A four-sided polygon is a parallelogram if and only if it has two sets of parallel sides.

    Rhombus: A four-sided polygon is a rhombus if and only if its sides are equal and has two pairs of congruent angles.

    Pentagon: A polygon is a pentagon if and only if it has 5 sides.

    Regular Pentagon: A polygon is a regular pentagon if and only if it has 5 equal sides.

    Trapezoid: A four-sided polygon is a Trapezoid if and only if it has two parallel sides and two pairs of congruent angles.

    Hexagon: A polygon is a hexagon if and only if it has 6 sides.

    Thank you for your help.

    Edit: I have avoided simply looking up the definitions on the internet so that I'm given the chance to reason my way to an answer, if possible.
     
    Last edited: Aug 13, 2011
  2. jcsd
  3. Aug 13, 2011 #2

    I like Serena

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    Hi Dembadon! :smile:

    Just a few comments on your statements (mostly nitpicking, but that is what math is about :wink:).

    In a rhombus the sides are not equal (that would be pretty weird!), but the lengths of the sides are equal.

    The two pairs of congruent angles are redundant in a rhombus (but not wrong).

    A pentagon with 5 sides of equal length does not have to be regular.
    Perhaps you could try to find an example?

    A trapezoid does not have to have two pairs of congruent angles.
    Perhaps you could look up the definition of a trapezoid?

    The rest looks good! :smile:
     
  4. Aug 13, 2011 #3

    Dembadon

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    Gold Member

    Hello!

    Thank you, ILS. Nitpicking is exactly what I was hoping for! I need, and want, to learn to be as precise as possible.
     
  5. Aug 13, 2011 #4

    I like Serena

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    Good!
    I've been tuning my nitpicking back in real life, since it usually mostly irritates people, but I believe it is invaluable in math and programming. :wink:
     
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