Question

what is the measure of (overarc{ac})? a. (150^circ) b. (120^circ) c. (30^circ) d. (90^circ)

Explanation:

Step1: Find arc DE measure

The central angle for $\widehat{DE}$ is $30^\circ$, so $m\widehat{DE}=30^\circ$.

Step2: Find arc AE measure

$\angle ABE$ is a straight angle ($180^\circ$), so $m\widehat{AE}=180^\circ$.

Step3: Find arc AD measure

$m\widehat{AD}=m\widehat{AE}-m\widehat{DE}-m\widehat{BD}$
$m\widehat{AD}=180^\circ-30^\circ-90^\circ=60^\circ$

Step4: Find arc AC measure

$\widehat{AC}$ is a semicircle minus $\widehat{AD}$? No, correct: $\widehat{AC} = \widehat{ABD} + \widehat{BC}$. $\widehat{ABD}=90^\circ+60^\circ=150^\circ$? Wait, simpler: The central angle for $\widehat{AC}$: $\angle ABC$ is supplementary to $\angle ABD$? No, $\widehat{AC} = 180^\circ + m\widehat{AD}$? No, better: The inscribed angle/central angle: $\widehat{AC}$ corresponds to the angle that is $180^\circ - 30^\circ = 150^\circ$? Wait, no, straight line at B: $\angle ABD=90^\circ$, $\angle DBE=30^\circ$, so $\angle ABE=180^\circ$. The arc $\widehat{AC}$: since $\widehat{BC}$ is $180^\circ - \widehat{BE}$, $\widehat{BE}=90^\circ+30^\circ=120^\circ$, so $\widehat{BC}=60^\circ$, then $\widehat{AC}=\widehat{AB}+\widehat{BC}=90^\circ+60^\circ=150^\circ$.

Answer:

A. $150^\circ$