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Topic: New Rule for Naming Polygons
Replies: 1   Last Post: Apr 29, 2000 2:47 PM

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 JIMMY Posts: 7 Registered: 12/6/04
New Rule for Naming Polygons
Posted: Apr 15, 2000 5:37 PM

In June 2003, I think it will be time to use a new rule for naming
polygons. Are you ready to hear it??

The first change is that the word "triangle" will be dropped
completely from our language. The 2 new words will be "trigon", for
equilateral triangles, and "trilateral", for any other triangle.

Also, there will be 2 new terms for special polygons with any number
of sides:

Regular: A regular polygon is a polygon with all sides congruent and
all angles congruent.

Semiregular: A semiregular polygon is a polygon that is not regular,
but whose vertexes made by its angles all lie on one circle of a given
size.

Note that using these definitions, triangles are always either regular
or semiregular. In other words, a regular 3-sided figure will be
called a trigon and any other 3-sided figure will be called a
trilateral. Polygons with 4 or more sides, however, can be regular,
semiregular, or neither. For naming 4-sided polygons, we will use all
the same terms used earlier, but change the definitions as follows:

Kite: A quadrilateral with 2 pairs of consecutive congruent sides, but
opposite sides not NECESSARILY congruent.

Trapezoid: A quadrilateral with AT LEAST one pair of parallel sides.

Isosceles trapezoid: A trapezoid whose BASE ANGLES are congruent.

Parallelogram: A trapezoid whose opposite angles are congruent.

Also, note the following theorem:

A trapezoid is also a semiregular quadrilateral if and only if it is
an isosceles trapezoid but not a square. (A square is the regular
quadrilateral.) So, we now have a new acceptable definition of an
isosceles trapezoid: a quadrilateral that is both a trapezoid and
either regular or semiregular. Also, we can extend this theorem to
saying that a quadrilateral is semiregular if and only if its opposite
angles add up to 180 degrees but is not a square and that a
quadrilateral is regular if and only if it is a square.

So now we have trigon and square for regular polygons with 3 and 4
sides, respecively, and trilateral and semiregular quadrilateral for
semiregular polygons with 3 and 4 sideds, respectively. Also, a
quadrilateral that is neither regular nor semiregular will be called a
"4-side". The word "quadrilateral" will be the only word retained in
our language that can be used to refer to polygons with a given number
(4) of sides that can be regular, semiregular, or neither.

The terms "pentagon", "quintilateral", and "5-side" will be used for
5-sided figures that are regular, semiregular, and neither,
respectively. The terms "hexagon", "sextilateral", and "6-side" will
be used for 6-sided figures that are regular, semiregular, and
neither, respectively.

Now, let me go from 3 to 12 sides, and make a list of all the
acceptable names for polygons that are regular (all sides are
congruent and all angles are congruent,) semiregular (vertexes all lie
on a circle but sides are not all congruent,) and neither.

3 trigon, trilateral
5 pentagon, quintilateral, 5-side
6 hexagon, sextilateral, 6-side
7 heptagon, septilateral, 7-side
8 octagon, octilateral, 8-side
9 enneagon, nonilateral, 9-side
10 decagon, decilateral, 10-side
11 hendecagon, undecilateral, 11-side
12 dodecagon, duodecilateral, 12-side
General regular polygon, semiregular polygon, polygon that is neither
regular nor semiregular

Other terms:

polygon: any number of sides
quadrilateral: 4 sides, can be regular, semiregular, or neither
kite: quadrilateral with 2 pairs of consecutive congruent sides
trapezoid: quadrilateral with at least one pair of parallel sides
isosceles trapezoid: trapezoid with base angles congruent
parallelogram: trapezoid with opposite angles congruent
rhombus: quadrilateral that is both a kite and a parallelogram
rectangle: trapezoid that is both an isosceles trapezoid and a
parallelogram

Note that "3-side" is not included because all 3-sided polygons are
either regular or semiregular.

That's it! Now we have 3 types of polygons (regular, semiregular,
neither) and 3 languages (Greek, Latin, English,) so we can think of
one as corresponding to the other in the following way. Can you PLEASE