Question

determining central and inscribed angles in circles
the ( mwidehat{wj} ) is
reason:

Explanation:

Step1: Recall the inscribed - angle theorem

The measure of an inscribed angle in a circle is half the measure of the central angle that subtends the same arc. The formula is \(m\angle=\frac{1}{2}m\overset{\frown}{arc}\), where \(m\angle\) is the measure of the inscribed angle and \(m\overset{\frown}{arc}\) is the measure of the arc.

Step2: Identify the inscribed angle

In the given circle, \(\angle WZJ\) is an inscribed angle that subtends arc \(WJ\). We are given that \(m\angle WZJ = 39.7^{\circ}\).

Step3: Use the formula to find the arc measure

By the inscribed - angle theorem, if \(m\angle WZJ=\frac{1}{2}m\overset{\frown}{WJ}\), then \(m\overset{\frown}{WJ}=2\times m\angle WZJ\).
Substitute \(m\angle WZJ = 39.7^{\circ}\) into the formula: \(m\overset{\frown}{WJ}=2\times39.7^{\circ}=79.4^{\circ}\).

Answer:

The \(m\overset{\frown}{WJ}\) is \(79.4^{\circ}\). Reason: The measure of an inscribed angle (\(\angle WZJ\)) is half the measure of the arc (\(\overset{\frown}{WJ}\)) it subtends. Using the formula \(m\overset{\frown}{arc}=2\times m\angle\) (where \(\angle\) is the inscribed angle), with \(m\angle = 39.7^{\circ}\), we get \(m\overset{\frown}{WJ}=2\times39.7^{\circ}=79.4^{\circ}\).