SOLUTION: How many sets of congruent triangles are there when you draw all diagonals of a regular pentagon and how do you know they are congruent?

There are 5 sets of congruent triangles.

I have lettered all 11 regions A through K.
I will indicate triangles by writing the regions
which they contain, such as A represents triangle
whis is just region A.  AB will represent the
triangle composed of regions A and B.  ABC will
represent the triangle composed of regions A,B,
and C.  ABDEI is the triangle composed of regions
A,B,C,D,E, and I.

1. These 5 triangles are congruent: A≅C≅G≅J≅K
2. These 5 triangles are congruent: B≅D≅F≅H≅I
3. These 10 triangles are congruent: 

AB≅BC≅CF≅FJ≅IJ≅IK≅HK≅GH≅GD≅AD
 
4. These 10 triangles are congruent: 

ABC≅ADG≅GHK≅KIJ≅CFJ≅DEF≅BEH≅DEI≅EFH≅BEI

5. These 5 triangles are congruent: ABDEI≅BCEFH≅DEFIJ≅BEHIK≅DEFGH

That makes 35 triangles.

They are congruent by the law of common sense of symmetrical
things. [Yes I know that's not a proof! :)  ]

It wouldn't be hard to prove any two of those congruent, but
all 35?  Forget it!  If you pick two that you want me to
prove congruent, I can help you do that, but proving that 
all pairs of congruent triangles above are really congruent 
is an unreasonable task that would take many hours.

Edwin