Answer: x = 10
Explanation
Let point A be at the center of the circle.
Minor arc DE = 120 degrees is subtended by central angle EAD, so angle EAD is also 120 degrees.
Notice how triangle EAD is isosceles (since AE and AD are congruent radii). Therefore, the congruent base angles are E and D as they are opposite the congruent radii.
Focusing on triangle EAD we can say...
E+A+D = 180
E+120+E = 180
2E = 180-120
2E = 60
E = 60/2
E = 30
Angles E and D of triangle EAD are 30 degrees each.
This will mean angle AED = 30, and,
(angle AED) + (angle DEF) = angle AEF
(30) + (angle DEF) = 90
angle DEF = 90-30
angle DEF = 60
Set this equal to the 7x-10 in the diagram (since this is also the measure of angle DEF)
7x-10 = 60
7x = 60+10
7x = 70
x = 70/7
x = 10 is the final answer.
