What is the fewest number of angularly unique isosceles triangles that can be pieced together to form a...

Congratulations to Mauro Fiorentini of Italy who was the only person
to find a solution to the Isosceles Triangles Puzzle.

I found a solution to this puzzle using fewer than 6 triangles, and
challenged the world to find a different solution.  Mauro was the only
one to send me a solution, and what do you know... it was different
than mine!  In lieu of $50 prize, Mauro has opted for merchandise.
"Jay... what do we have for our contestant this evening?"

   "Well, Evan, it's a collection of mathematical books by Martin
Gardner!" [Applause]

Both solutions work only for a specific set of angles, so it is just
possible that there are EXACTLY TWO solutions to this puzzle, but I'm
afraid our prize warehouse is empty, so if anyone finds yet a third
solution, you'll have to settle for fame and glory, only.

Both solutions share one common triangle.  Here is a hint ROT13'd:

     bar natyr vf avargl qrterrf

Also, Mauro's solution contains two triangles that are angularly-
unique, but have the same area; whereas, each triangle in my solution
has a different area.  Since "unique area" was not a requirement
Mauro's solution is perfectly valid.  Is there more than one solution
that is angular- AND area-unique?  Who knows??

Here is one way how to cut a square into 5 angularly-unique isosceles
triangles:

+--------------------------------+
| \                            A |
|B  \                            |
|     \                          |  A : 90     45     45
|       \                        |  B : 45     67.5   67.5
|         \                      |  C : 112.5  33.75  33.75
|           \                    |  D : 135    22.5   22.5
|             \                  |  E : 157.5  11.25  11.25
|               \                |
|                 \              | Each triangle is labeled with
|                   \            | its letter placed closest to the
|                     \          | triangle's apex (unique angle).
|                 ___/C \        |
|             ___/        \      | Each triangle's area is unique.
|         ___/.....         \    |
|     ___/D\___     .....     \  |
|____/         \____      ......\|
+-------------------\E-----------+

Here is "the other" way:

+-------------------------------+
|\                            A |
|B \                            |
|    \                          |  A : 90     45     45
|      \                        |  B : 45     67.5   67.5
|        \                      |  C : 112.5  33.75  33.75
|          \                    |  D : 22.5   78.75  78.75
|            \                  |  E : 157.5  11.25  11.25
|              \                |
|                \              | Each triangle is labeled with
|            _____/\            | its letter placed closest to the
|      _____/      C \          | triangle's apex (unique angle).
|_____/                \        |
|     \_____             \      | D and E have the same area.
|          D\______        \    |
|      _____/   E  \_____    \  |
|_____/                  \_____\|
+-------------------------------+

Several folks were more interested in my code than in the puzzle.  I
merely concatenated all of the digits of the unique angles of each
triangle and then added PI digit-for-digit to produce the final code.
To decode, subtract PI digit-for-digit from the code as shown:

  471672776613569      code
- 314159265358979    - digits of PI
=================
  157513511254590 -> 157.5, 135, 112.5, 45, 90
                        E    D     C    B   A   (For first solution)

-- Evan Whitney <evan@scd.hp.com>  (August 1998)

[ Editor: the hint decodes to "one angle is ninety degrees".  The
original challenge, posted earlier with the code number, is not in the
collected material. ]