Question

question 10 of 10 what is the measure of $widehat{ac}$?

Explanation:

Step1: Recall arc - angle relationship

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

Step2: Find measure of arc $DE$

Since $\angle DBE = 90^{\circ}$ and it is an inscribed angle, the measure of arc $DE$ is $180^{\circ}$ (because the central angle corresponding to arc $DE$ is $180^{\circ}$ as $\angle DBE$ is inscribed and the central - angle is twice the inscribed angle).

Step3: Find measure of arc $AD$

We know that $\angle ADB = 42^{\circ}$, and by the inscribed - angle theorem, the measure of arc $AB$ is $84^{\circ}$ (since the measure of an inscribed angle is half the measure of its intercepted arc, so the arc $AB = 2\times42^{\circ}=84^{\circ}$).

Step4: Calculate measure of arc $AC$

The measure of arc $AC$ is the sum of the measures of arc $AB$ and arc $BC$. Arc $BC = 90^{\circ}$ (because $\angle ABC = 90^{\circ}$ and the central - angle corresponding to arc $BC$ is $180^{\circ}$ for the right - angled inscribed angle $\angle ABC$). The measure of arc $AB = 48^{\circ}$ (from the $42^{\circ}$ inscribed angle $\angle ADB$, arc $AB = 2\times42^{\circ}=84^{\circ}$, but we made a wrong start above. Let's start over.
The measure of arc $DE=84^{\circ}$ (since inscribed angle $\angle DBE = 42^{\circ}$, arc $DE = 2\times42^{\circ}=84^{\circ}$). The measure of arc $AB = 90^{\circ}$ (because $\angle ABD = 90^{\circ}$ and the central - angle corresponding to arc $AB$ is $180^{\circ}$ for the right - angled inscribed angle $\angle ABD$).
The measure of arc $AC=138^{\circ}$ because arc $AC=$ arc $AB +$ arc $BC$, and arc $AB = 90^{\circ}$, arc $BC = 48^{\circ}$ (since arc $DE = 84^{\circ}$, and the remaining part of the semi - circle from $B$ to $C$ is $48^{\circ}$). The measure of arc $AC=90^{\circ}+ 48^{\circ}=138^{\circ}$.

Answer:

A. $138^{\circ}$