Question

Can someone help me pls

Answer 1

Answer:

$\boxed{ m \overset{\huge\frown}{EFC} \: = \: 237°}$

Explanation:

Assuming that $\overline{CF}$ is a diameter, and that the angle of an arc subtending the central angle is congruent, $m \overset{\Large\frown}{CGF} \: = \: 180°$.

And according to the arc addition postulate, $\overset{\Large\frown}{CG} + \overset{\Large\frown}{GF} \: = \: \overset{\huge\frown}{CGF}$.

Using the substitution property:

$\overset{\Large\frown}{CG} + \overset{\Large\frown}{GF} \: = \: 180°$.

Given that $m\overset{\Large\frown}{CG} \: = \: 140°$.

Using the substitution property and that the measures apply to the arc addition postulate, $140° + m\overset{\Large\frown}{GF} \: = \: 180°$.

$140° – \: 140° + m\overset{\Large\frown}{GF} \: = \: 180° – \: 140°$.

$m\overset{\Large\frown}{GF} \: = \: 40°$.

Since $\overset{\Large\frown}{GF}$ and $\overset{\Large\frown}{CD}$ are subtending the center of the circle, they form vertical angles and are therefore congruent.

$\overset{\Large\frown}{GF} \: \cong \: \overset{\Large\frown}{CD}$

According to the definition of a vertical angle, $m \overset{\Large\frown}{CD} \: = \: 40°$.

Using the arc addition postulate again,

$\overset{\Large\frown}{CD} + \overset{\Large\frown}{DE} + \overset{\Large\frown}{EF} \: = \:\overset{\huge\frown}{CDF}/\overset{\huge\frown}{CEF}$

Using substitution, as well as how we are given that the central angle subtended by $\overset{\Large\frown}{DE}$ = 83°, we can find that:

$40° + 83° + \overset{\Large\frown}{EF} \: = \:\overset{\huge\frown}{CDF}/\overset{\huge\frown}{CEF}$