Nice work, Logitech!

One important piece is to recognize that triangles ACB and BCD are similar triangles.

Check out my DS question (cars and students) - I'd love to see your assessment.

The diagram in the question is actually related to one of the most famous diagrams in geometry - the one Euclid used to prove the Pythagorean Theorem. The early proofs of Pythagoras relied on the fact that you get three similar triangles when you draw, as in the diagram above, a height connecting the hypotenuse to the vertex with the 90 degree angle. Because we get similar triangles in such a picture, we really don't need much additional information to find all the lengths here, as logitech observed above. One other useful takeaway: it's generally worthwhile looking for similar triangles in any diagram where one triangle contains another, since when this happens, the two triangles are certain to share at least one angle.

My only suggestion, from a question design point of view, would be to remove the contradiction between the two statements - for example, to make Statement 1 read "CB = 3.5", so that it's logically possible to consider both statements together (using the first statement, each triangle is similar to a 3-4-5 triangle, while using the second, each triangle is a 30-60-90 triangle). Of course, the real GMAT always ensures the two statements are logically consistent.

Nice call, Ian.
I forgot to ensure consistency.