Thread: 2-manifold question

So i have this question that had me stumped but I may have got it.
I have a polyhedral surface which is formed from the following shaded triangular region, by gluing the edges of the inner tringle to those of the outer triangle (image 1).

Now the question is saying that the triangle is glued as follows (image 2). It then goes on to say that if i make a suitable cut, or cuts, the surface can be obtained by gluing together the edges of an octagon. I then have to write the corresponding word (abcda'b'c'd' for example) hence showing that my surface is homeomorphic to a torus.

Now I think I have the cut in the right place (image 3) which would give me the octagon with word abcdc'b'a'd' (image 4). Firstly is this correct?
I then go on to show that
abcdc'b'a'd' = aede'a'd' = ede'aa'd' = ede'd' = aba'b'
which is a torus. Is my working here right? I have similar questions on this and even though I got the answer I was looking for it would be great to know whether my working is right or completely wrong (as the case may be!)

Thanks in advance!