MATHEMATICS AND ORIGAMI

Mathematics and Origami

1.3.4 VERTICAL ANGLES

1.3.5 SUM OF THE ANGLES OF A TRIANGLE

Let ́s produce the following folds in ∆ABC:
1- CD: C → C ; AB → AB
2- FG: C → D
As result we have:

ED CD

2

1

= ; FA CA

2

1

= ; GB CB

2

1

=

Besides, because of the symmetry, it is:
FC=FD ; GC=GD

which proves that ∆AFD and ∆DGB are isosceles and therefore:
Ang.FDA = Ang.FAD ; Ang.GDB = Ang.GBD

The straight angle in D can be expressed as:
180 = Ang.ADF + Ang. FDG +Ang. GDB
or its equivalent:
180 = Ang.CAB + Ang. ACB +Ang. CBA
This proves that the three angles in a triangle add up to 180º.

C A

F

A E D

F D

A

F E D

B

B C

O

a

p

b

Let a and b, lines meeting at O.

If we produce the fold p: O → O ; b →a

angles α will be equal, respectively, because of symme-

try. Therefore vertical angles 2α and 2α will also be

equal

C

F E G

A D B

F E G

A D B