MATHEMATICS AND ORIGAMI

Mathematics and Origami

From the above we can ́t jump to the conclusion that the pentagon is a regular one: we
know of equalities but not of values. Last equalities are also in pentagon (5) that is not regular.

Let ́s note now in (3) the four trapeziums

ABCE ; ABDE ; ACDE ; ABCD

Incidentally, when fig. 3 is unfolded, those four trapeziums appear as shown in fig. 6.

Trapeziums ABCE and ABDE are isosceles for both have:

• Their parallel bases distant h from each other.


• Big angles, congruent.


• Skew sides and small base, congruent.
  Former conditions lead to trapeziums congruency. Then their diagonals should be con-
  gruent too:
  AC = AD
  Let ́s have a look now to the trapeziums ACDE and ABCD that in turn have equal:


• The altitude h as the distance between their parallel bases.


• Their great bases AD = AC


• Their small bases BC = ED


• The small base congruent with a skew side: BC = AB = DE = AE


• The great angle formed by the small base and one skew side: Ang.B = Ang. E
  Consequently these trapeziums are congruent with each other, and also with the other
  pair of former trapeziums.
  Therefore Ang. C = Ang. D, and congruent with the other four angles of the pentagon.
  As CD = l, pentagon ABCDE is a regular one for all its sides and angles are, respectively, con-
  gruent.

A

B

C D

E

4

C

B ́

5

D

E ́

A

C ́ D ́

B A C D

EA

B

CDE

h^6