Question

what is the measure of \\( \overarc{bc} \\)?
enter your answer in the box.
\\( \square ^ { \circ } \\)

Explanation:

Step1: Recall the property of a circle

The measure of an arc is related to the central angle. But here, we assume \(BD\) is a diameter (since it's a common setup in such problems). The sum of angles around a point on a line (diameter) is \(180^{\circ}\).

Step2: Calculate the measure of arc \(BC\)

If \(BD\) is a diameter, the measure of the arc \(BDC\) (semic - circle) is \(180^{\circ}\). Let the measure of arc \(BC\) be \(x\). We know that the inscribed - angle - related formula (if we consider the relationship between the angles subtended by the arcs). But more simply, if we assume the angle \(\angle BDC = 65^{\circ}\) and using the fact that the measure of an arc is equal to the measure of the central angle that subtends it. If \(BD\) is a diameter (central angle for semic - circle \(= 180^{\circ}\)), and if we assume the angle at \(D\) (if it's a central angle, but in a circle, the measure of an arc \(BC\) is \(180^{\circ}-\text{(measure of arc }BD\text{)}\). Wait, no, if we consider the fact that the sum of arcs in a circle: If \(BD\) is a diameter (arc \(BD = 180^{\circ}\)), and if we assume that the angle \(\angle BDC\) is a central angle (a wrong assumption, actually, the measure of an arc is equal to the measure of the central angle that subtends it. Let's re - do.

The measure of an arc \(BC\): We know that the sum of arcs in a circle. If we assume \(BD\) is a diameter (arc \(BD\) is \(180^{\circ}\)). Let the measure of arc \(BC=x\). If we consider the fact that the angle \(\angle BDC\) (if it's an inscribed angle, no, wait, no, in a circle, the measure of an arc is equal to the measure of the central angle that subtends it. Wait, another approach: The total degrees in a circle is \(360^{\circ}\), but if \(BD\) is a diameter (a straight line through the center, so the arc \(BD\) is \(180^{\circ}\)). If we assume that the given \(65^{\circ}\) is the measure of the central angle for arc \(DC\). Then the measure of arc \(BC\) is \(180^{\circ}- 65^{\circ}\)

$$180^{\circ}-65^{\circ}=115^{\circ}$$

Answer:

\(115\)