[psyang]

We have a winner!

It's a bit of a trial/error approach, but it definitely works.

I think you hit on all the key concepts for the solution. It can be more elegantly summed up as:

Spoiler!

Start with an equilateral triangle of length 9.

Recognizing the internal angles are 120, and the complementary angles are therefore 60, it's a matter of chopping off 3 equilateral triangles off the original.

Originally Posted by SebC

Spoiler!

I was going to go with no, but then I tested it (sort of cheating I guess) and I found one.

So here's how it works: for all interior angles to be equal, the interior angles must all be 120 degrees. So you have 6 across the top. Attach 1 on the left. Because that's a 60 degree triangle the width of the polygon added for that piece is 0.5. Stick the 2 on the right, horizontal distance added 1.0. Total 7.5. Connect the 5 to the one. 2.5 horizontal travelled. 4 to the 2. Width there is 2.0. Throw the 3 in the middle of the bottom, total horizontal of the bottom three segments is 7.5 again, so they match. Looking at the height of the polygon, you've got 5+1 on one side and 4+2 as the length of your slopes. Whether it is going left or right doesn't matter, the height scales with the total length so the heights of both sides are equal (height is 6*sqrt(3)/2=3*sqrt(3)).