## Task A-2.3.

On a circle, five points are given as shown in the figure. Segments $\overline{A C}$ and $\overline{B E}$ intersect at the center of the circle. If $\varangle D A C=37^{\circ}$ and $\varangle E B D=28^{\circ}$, determine the angle $\varangle A C E$.

![](https://cdn.mathpix.com/cropped/2024_05_30_4cdaff2107ef283b1d94g-08.jpg?height=514&width=551&top_left_y=1159&top_left_x=1141)
## First solution.

Notice that $\varangle E C D=\varangle E B D=28^{\circ}$ because these are inscribed angles over the chord $\overline{D E}$.

![](https://cdn.mathpix.com/cropped/2024_05_30_4cdaff2107ef283b1d94g-08.jpg?height=611&width=643&top_left_y=1865&top_left_x=638)

Since $\overline{A C}$ is the diameter, $\varangle A D C=90^{\circ}$.

From triangle $A C D$, we conclude that

$$
\varangle E C A=180^{\circ}-\varangle A D C-\varangle C A D-\varangle D C E=90^{\circ}-37^{\circ}-28^{\circ}=25^{\circ}
$$