in circle e, $overline{ac}$ and $overline{bd}$ are diameters. angle bca measures 53°. what is the measure of…

Answer:

C. 106°

Explanation:

Step1: Recall inscribed - angle theorem

The measure of an inscribed angle is half the measure of its intercepted arc. Angle BCA is an inscribed angle that intercepts arc AB. Given (\angle BCA = 53^{\circ}), then the measure of arc AB is (2\times\angle BCA). So, (m\overset{\frown}{AB}=2\times53^{\circ}=106^{\circ}).

Step2: Use property of vertical angles and arcs

Since (\overline{AC}) and (\overline{BD}) are diameters, (\angle AEB) and (\angle DEC) are vertical angles, and (\angle BEC) and (\angle AED) are vertical angles. Also, the sum of the measures of the arcs of a circle is (360^{\circ}). Arc AD and arc BC are vertical - arc pairs. Arc AB and arc AD are adjacent arcs. Since (\angle BCA) intercepts arc AB and we want to find arc AD, and we know that the measure of arc AB is (106^{\circ}), and the measure of arc AD is equal to the measure of arc AB (vertically opposite arcs formed by the intersection of two diameters in a circle have equal measures). So, (m\overset{\frown}{AD}=106^{\circ}).