Question

finding an arc measure
\\(\overline{gd}\\) is a diameter of circle c.
what is \\(m\widehat{ceb}\\)?
\\(\circ\\) \\(90^\circ\\)
\\(\circ\\) \\(180^\circ\\)
\\(\circ\\) \\(219^\circ\\)
\\(\circ\\) \\(231^\circ\\)

Explanation:

Step1: Identify straight angle (GD)

Since $\overline{GD}$ is a diameter, $m\angle GCD = 180^\circ$.

Step2: Calculate $m\angle ECD$ complement

The right angle gives $m\angle FCD = 90^\circ$, so $m\angle ECF = 90^\circ - 39^\circ = 51^\circ$.

Step3: Sum angles for arc $\overparen{CEB}$

Add $m\angle ECF$, $m\angle FCG$, $m\angle GCA$, $m\angle ACB$. First, $m\angle GCA + m\angle ACB = 180^\circ - 51^\circ - 39^\circ = 90^\circ$. Then:
$$m\overparen{CEB} = 51^\circ + 51^\circ + 90^\circ + 39^\circ = 231^\circ$$

Answer:

231°