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None, as far as I can see. Which, given that I cannot see any triangle, is not saying much!
I assume that points BC is a straight line and that D is the point where the line DE intersects BC. If this is so, and CDE is 55 degress, then BDE is 125 degrees.
10 (x 6 if no repetition is allowed but abc is not regarded as the same as acb), 125 if it is. ABC/ABD/ABE/ACD/ACE/ADE/BCD/BCE/BDE/CDE
20: abc, abd, abe, abf, acd, ace, acf, ade, adf, aef, bcd, bce, bcf, bde, bdf, bef, cde, cdf, cef, def.
If you can only take one of each letter, it would be 19: abc, abd, abe, abf, acd, ace,acf,ade,adf,aef, bcd, bce, bcf, bde, bdf, bef, cde, cdf ,def.