Question

consider circle e. \\( \overleftrightarrow { c d } \\) is the perpendicular bisector of segment \\( \overline { a b } \\).
what is the measure of the arc intercepted by \\( \angle c \\)?

Explanation:

Step1: Recall the property of inscribed angle

The measure of an inscribed angle is half the measure of its intercepted arc. Let the intercepted arc be \( \overset{\frown}{AB}\).

Step2: Determine the measure of \( \angle AEB\)

Since \(CD\) is the perpendicular bisector of \(AB\) and \(E\) is the center of the circle, \( \angle AEB = 90^{\circ}\) (perpendicular lines form a \(90^{\circ}\) angle).

Step3: Find the measure of \( \angle C\)

\( \angle C=\frac{1}{2}\angle AEB\) (inscribed angle theorem). Substituting \( \angle AEB = 90^{\circ}\), we get \( \angle C = 45^{\circ}\). Then the measure of the arc intercepted by \( \angle C\) (which is \( \overset{\frown}{AB}\)) is \(90^{\circ}\) (because \( \angle C=\frac{1}{2}\text{measure of } \overset{\frown}{AB}\)).

Answer:

\(90^{\circ}\)