Geometry
posted by Aaron .
I have a circle with a Tangent line, DE, running along it and connecting with another line, FC, which runs through the center of the circle to the other side.I have connected the center of the circle with point E on the edge of the circle with a radius. I have labeled the center of the circle O.
Angle D equals 40 degrees. Angle DEO equals 90 degrees. Angle FOE equals 50 degrees.
What is arc measure EF?

Don't know about anybody else, but I find the description of your diagram very confusing.
"I have a circle with a Tangent line, DE, running along it and connecting with another line, FC, which runs through the center of the circle to the other side."
Is E the contact point?
Are F and C on the circle?
"Angle D equals 40 degrees"
What are the arms of angle D?
Is D the intersection of line FC extended?
Finally you ask for the measure of the arc EF, but you give only angles and no measure of any sides.
the arc length will be a function of the radius of the circle, and since we have no lengths given at all .... ????? 
E is the contact point.
F and C are on the circle and 180 degrees from each other.
Angle D is EDF.
finally, I need the angle measure of Arc EF 
you said "finally, I need the angle measure of Arc EF "
I assume you meant the central angle subtended by arc EF ?
But didn't you say that angle FOE = 50ยบ ? 
oh...
wow, thx.