let

d = the measure of angle D

e = the measure of angle E

f = the measure of angle F

the sum of the measure of angle D and angle E is 110:

d + e = 110

e = 110 - d

the sum of the measures of angle E and angle F is 150:

e + f = 150

e = 150 - f

add e = 110 - d and e = 150 - f:

2e = 110 - d + 150 - f

2e = 260 - (d + f)

e = (1/2)(260 - (d + f))

since they are angles in a triangle:

d + e + f = 180

plug e = (1/2)(260 - (d + f)) into the last equation:

d + (1/2)(260 - (d + f)) + f = 180

130 + (1/2)(d + f) = 180

(1/2)(d + f) = 180 - 130

(1/2)(d + f) = 50

d + f = 50*2

d + f = 100

the measures of angle D and angle F is 100.