Question

in circle o, ac and be are diameters. the measure of arc dc is 50°. what is the measure of arc ebc? 220° 140° 90° 40°

Explanation:

Step1: Recall circle - arc properties

The sum of the measures of the arcs of a circle is 360°. Since \(AC\) and \(BE\) are diameters, \(\angle AOB=\angle DOE = 90^{\circ}\) and the measure of arc \(DC = 50^{\circ}\).

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

The measure of arc \(EBC\) can be found by considering the whole - circle and the non - included arcs. The non - included arcs are arc \(ED\) and arc \(DC\). Arc \(ED\) corresponds to the central angle \(\angle DOE=90^{\circ}\) and arc \(DC = 50^{\circ}\). So the measure of arc \(EBC=360^{\circ}-(90^{\circ}+50^{\circ})\).
\[360-(90 + 50)=360 - 140=220^{\circ}\]

Answer:

\(220^{\circ}\)