Question

in circle c, diameter \\(\overline{fd}\\) is perpendicular to chord \\(\overline{ab}\\).
select all of the arcs and angles that have the same measure as \\(\overarc{ad}\\).
\\(\angle dcb\\) \\(\overarc{bg}\\) \\(\angle cfa\\) \\(\angle agb\\)

Explanation:

Step1: Use perpendicular diameter theorem

Since diameter $\overline{FD} \perp \overline{AB}$, it bisects $\overline{AB}$ and the arcs $\overset{\frown}{AB}$, so $\overset{\frown}{AD} = \overset{\frown}{BD}$. Also, the central angle over an arc equals the arc's measure: $m\angle DCB = m\overset{\frown}{BD} = m\overset{\frown}{AD}$.

Step2: Relate inscribed angles to arcs

An inscribed angle is half its subtended arc: $m\angle CFA = \frac{1}{2}m\overset{\frown}{AD}$, so it does not match. $m\angle AGB = \frac{1}{2}m\overset{\frown}{AB}$, which is not equal to $m\overset{\frown}{AD}$. $\overset{\frown}{BG}$ has no given relation to $\overset{\frown}{AD}$.

Step3: Verify matching measure

Only $\angle DCB$ has the same measure as $\overset{\frown}{AD}$, as the central angle is congruent to its intercepted arc, and the perpendicular diameter makes $\overset{\frown}{AD}=\overset{\frown}{BD}$, so $\angle DCB$ (central angle for $\overset{\frown}{BD}$) equals $m\overset{\frown}{AD}$.

Answer:

$\angle DCB$