In the figure above, quadrilateral ABCD is a parallelogram. Let x represent the measure of angle GBF, y represent the measure of angle CBE, and z represent the measure of angle BCE.

Question

Answer 1

Here's another way to get the same result as Donald W.:

GBF is angle x, DBE is also angle x (vertical angles). We're given BDE as 50°. Total of angles within triangle DBE is 180°. Angle x is therefor 180 - 50 - 90 = 40°. Now that we know x, we can find y using the fact that supplementary angle FBE is 180°. 180 - 40 - 105 = 35° That's angle y. Now can use triangle BCE to find angle z (BCE). 180 - 35 - 90 = 55° for angle z. As Sal Khan would say "and we're done".

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Answer 2

I'm assuming the question is asking us to solve for x, y, and z. First, we know that ∠CBD is 75 degrees because it is supplemental to ∠CBG. We can then deduce that ∠BCE is 55 degrees because it is the third angle for triangle CBD. From that, we can deduce that ∠CBE is 35 degrees because it is the third angle for triangle CBE. Lastly, we can deduce that ∠GBF is 40 degrees because it is supplemental to ∠GBE.

So x is 40 degrees, y is 35 degrees, and z is 55 degrees.