Question

in circle a, bd is a diameter, and the measure of cb is 48°. what is the measure of ∠dbc?

Explanation:

Step1: Recall inscribed - angle theorem

The measure of an inscribed angle is half the measure of its intercepted arc.

Step2: Identify the intercepted arc

The inscribed angle \(\angle DBC\) intercepts arc \(CD\). The measure of arc \(CB\) is given as \(48^{\circ}\). Since \(BD\) is a diameter, the measure of arc \(CDB\) is \(180^{\circ}\), so the measure of arc \(CD=180^{\circ}- 48^{\circ}=132^{\circ}\).

Step3: Calculate the measure of \(\angle DBC\)

By the inscribed - angle theorem, if \(\theta\) is the measure of the inscribed angle and \(s\) is the measure of the intercepted arc, then \(\theta=\frac{1}{2}s\). For \(\angle DBC\) with intercepted arc \(CD\) of measure \(132^{\circ}\), we have \(\angle DBC = \frac{1}{2}\times132^{\circ}=66^{\circ}\).

Answer:

\(66^{\circ}\)