Question

circle c is shown with diameters be and ad. the measure of angle bac is 35°. what is the measure, in degrees, of de? circle t is shown, where the measure of angle rsv is 23°. what is the measure of arc rv?

Explanation:

Step1: Recall the inscribed - angle theorem

The measure of an inscribed angle is half the measure of the intercepted arc. In circle \(C\), \(\angle BAC\) is an inscribed angle and \(\overset{\frown}{BC}\) is the intercepted arc. So \(m\overset{\frown}{BC}=2m\angle BAC\).
Since \(m\angle BAC = 35^{\circ}\), then \(m\overset{\frown}{BC}=2\times35^{\circ}=70^{\circ}\).

Step2: Use the property of vertical angles and arcs

\(\angle BAC\) and \(\angle ECD\) are vertical angles. So \(\angle ECD=\angle BAC = 35^{\circ}\). Also, \(\angle ECD\) is an inscribed angle and \(\overset{\frown}{DE}\) is the intercepted arc.
By the inscribed - angle theorem, \(m\overset{\frown}{DE}=2m\angle ECD\).
Since \(m\angle ECD = 35^{\circ}\), then \(m\overset{\frown}{DE}=70^{\circ}\).

For the second part:

Step1: Recall the relationship between a central angle and an arc

The measure of an arc is equal to the measure of its central angle. In circle \(T\), if \(\angle RSV\) is an inscribed angle and the central angle corresponding to arc \(\overset{\frown}{RV}\) is \(\angle RTV\).
The measure of an inscribed angle \(\theta_{i}\) and the central angle \(\theta_{c}\) that subtends the same arc are related by \(\theta_{i}=\frac{1}{2}\theta_{c}\) when the vertex of the inscribed angle is on the circle.
We know that the measure of the whole - circle is \(360^{\circ}\). The non - arc \(\overset{\frown}{RV}\) (the major arc) subtends an inscribed angle \(\angle RSV = 23^{\circ}\). The central angle corresponding to the non - arc \(\overset{\frown}{RV}\) is \(2\times23^{\circ}=46^{\circ}\).

Step2: Calculate the measure of arc \(\overset{\frown}{RV}\)

The measure of arc \(\overset{\frown}{RV}=360^{\circ}-46^{\circ}=314^{\circ}\)

Answer:

The measure of \(\overset{\frown}{DE}\) is \(70^{\circ}\) and the measure of arc \(\overset{\frown}{RV}\) is \(314^{\circ}\)