Question

chords $overline{de}$, $overline{ef}$, and $overline{fg}$ are congruent.
determine the following measures.
$moverset{\frown}{fg} = square^circ$
$mangle ecf = square^circ$
$\angle fcg = 56.5^circ$
$\angle ecd = 67^circ$
$\angle dch = 75^circ$

Explanation:

Step1: Relate chord congruence to arcs

In a circle, congruent chords correspond to congruent arcs. Since $\overline{DE} \cong \overline{EF} \cong \overline{FG}$, their corresponding arcs $\overset{\frown}{DE}$, $\overset{\frown}{EF}$, and $\overset{\frown}{FG}$ are congruent.

Step2: Find $\angle ECD$ measure

The central angle for $\overset{\frown}{DE}$ is $\angle ECD$. The sum of central angles around point $C$ is $360^\circ$. Calculate $\angle ECD$:
$$\angle ECD = \frac{360^\circ - 67^\circ - 56.5^\circ - 75^\circ}{2} = 80.75^\circ$$

Step3: Get $m\overset{\frown}{FG}$

Since $\overset{\frown}{FG} \cong \overset{\frown}{DE}$, $m\overset{\frown}{FG} = \angle ECD = 80.75^\circ$.

Step4: Identify $m\angle ECF$

$\angle ECF$ is given directly in the diagram as the central angle for $\overset{\frown}{EF}$:
$$m\angle ECF = 67^\circ$$

Answer:

$m\overset{\frown}{FG} = 80.75^\circ$
$m\angle ECF = 67^\circ$