Find the measure of angle "a" in the diagram.

This problem is known as Langley's Problem. It first appeared as 'A Problem', E M Langley, Mathematical Note 644, Mathematical Gazette, Volume 11, p. 173, 1922.

PDF Handout by Tom Rike (Berkeley Math Circle Handouts). An animated solution. Also see Adventitious Triangles and Langley type problems for an interesting discussion.