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.
Given that ABCD is a parallelogram, we know that angle ABC is equal to angle CDA, and angle BCD is equal to angle DAB. Therefore, angle EBC is equal to angle ECD.
Since angles ABC and CDA form a straight line, we have:
angle CDA = 180 - angle ABC
Since angle ABC is equal to angle CDA, we can substitute and solve:
angle CDA = 180 - angle CDA
2(angle CDA) = 180
angle CDA = 90
Since angle CDA is 90 degrees, angle BCE is also 90 degrees by the alternate interior angles theorem.
Therefore, z = 90.
Since angle EBC is equal to angle ECD, we have:
angle ECD = 180 - angle EBC
Again, substituting and solving:
angle EBC = 180 - angle EBC
2(angle EBC) = 180
angle EBC = 90
Since angle EBC is 90 degrees, angle CBE is also 90 degrees by the alternate interior angles theorem.
Therefore, y = 90.
Since angles ABC and CBE form a straight line, we have:
angle CBE = 180 - angle ABC
Therefore, angle ABC is equal to 90.
Since angles ABF and GBE are corresponding angles, we have:
angle GBF = angle ABF = 90
Therefore, x = 90.