Answer:
[tex] \boxed{ m \overset{\huge\frown}{EFC} \: = \: 237°} [/tex]
Explanation:
Assuming that [tex] \overline{CF} [/tex] is a diameter, and that the angle of an arc subtending the central angle is congruent, [tex] m \overset{\Large\frown}{CGF} \: = \: 180° [/tex].
And according to the arc addition postulate, [tex] \overset{\Large\frown}{CG} + \overset{\Large\frown}{GF} \: = \: \overset{\huge\frown}{CGF} [/tex].
Using the substitution property:
[tex] \overset{\Large\frown}{CG} + \overset{\Large\frown}{GF} \: = \: 180° [/tex].
Given that [tex] m\overset{\Large\frown}{CG} \: = \: 140° [/tex].
Using the substitution property and that the measures apply to the arc addition postulate, [tex] 140° + m\overset{\Large\frown}{GF} \: = \: 180° [/tex].
[tex] 140° – \: 140° + m\overset{\Large\frown}{GF} \: = \: 180° – \: 140° [/tex].
[tex] m\overset{\Large\frown}{GF} \: = \: 40° [/tex].
Since [tex] \overset{\Large\frown}{GF} [/tex] and [tex] \overset{\Large\frown}{CD} [/tex] are subtending the center of the circle, they form vertical angles and are therefore congruent.
[tex] \overset{\Large\frown}{GF} \: \cong \: \overset{\Large\frown}{CD} [/tex]
According to the definition of a vertical angle, [tex] m \overset{\Large\frown}{CD} \: = \: 40° [/tex].
Using the arc addition postulate again,
[tex] \overset{\Large\frown}{CD} + \overset{\Large\frown}{DE} + \overset{\Large\frown}{EF} \: = \:\overset{\huge\frown}{CDF}/\overset{\huge\frown}{CEF} [/tex]
Using substitution, as well as how we are given that the central angle subtended by [tex] \overset{\Large\frown}{DE}[/tex] = 83°, we can find that:
[tex] 40° + 83° + \overset{\Large\frown}{EF} \: = \:\overset{\huge\frown}{CDF}/\overset{\huge\frown}{CEF} [/tex]
