Problem Solving

The attached image is a problem that has been bothering me for a little while... below is the solution but after reading it I think the diagram is a little off... particularly the location of the 90-x angle. Any one else see this or am I missing something?

Based on the provided diagram I am seeing that the sum of the interior angles of triangle DEC is:

180 = 90-x+90+angle E
180 = -x+180+angle E
0=-x+angle E
x = angle E

Since CD is opposite angle E, and angle E =x, CD must be equal to BC (=5).

Based on the diagram isn't this the correct solution?


Below is the provided solution....


[spoiler]Applying The Pythagorean Theorem to the right triangle ABC yields
BC2 + AC2 = AB2
52 + AC2 = 102 given that AB = 10 and BC = 5 (from the figure)
AC2 = 102 - 52 = 100 - 25 = 75
Square rooting yields AC = sqrt(75) = sqrt(25)sqrt(3) =5 sqrt(3.)

Hence, the sides opposite angles measuring x° (A in triangle ABC) and 90° - x° (B in triangle ABC) are in the ratio
5 : 5 sqrt(3) = 1 : sqrt(3)

Similarly, in triangle ECD, the ratio of the sides opposite the angles E (measuring x°) and D (measuring 90° - x°)
must also be 1 : sqrt(3)

Hence, we have
CD/EC = 1 : sqrt(3)
CD/(5 + 5) = 1 : sqrt(3)
CD = 10 /sqrt(3)
The answer is (C).[/spoiler]