In the given figure, $A \widehat{O} D$ and $B \widehat{O} Y$ are right angles, and the measure of $D \widehat{O} Y$ is between $40^{\circ}$ and $50^{\circ}$. Additionally, points $C$ and $Y$ are on line $r$, while $D$ and $E$ are on line $s$. The possible value for the measure of $A \widehat{O} C$ is between

(a) $30^{\circ}$ and $40^{\circ}$;

(b) $40^{\circ}$ and $50^{\circ}$;

(c) $50^{\circ}$ and $60^{\circ}$;

(d) $40^{\circ}$ and $60^{\circ}$ or

(e) cannot be determined.

![](https://cdn.mathpix.com/cropped/2024_05_01_666a1aec7e447b0a6cbeg-032.jpg?height=323&width=403&top_left_y=1877&top_left_x=1432)
The correct answer is (b).

We have $A \widehat{O} C+C \widehat{O} E=90^{\circ}$ and $C \widehat{O} E=D \widehat{O} Y$. Therefore, $A \widehat{O} C=90^{\circ}-D \widehat{O} Y$. Since $D \widehat{O} Y$ is between $40^{\circ}$ and $50^{\circ}$, it follows that $A \widehat{O} C$ is between $90^{\circ}-50^{\circ}=40^{\circ}$ and $90^{\circ}-40^{\circ}=50^{\circ}$.