In the following figure, segment $AE$ is tangent to circle $c$ at $A$. Segment $AD$ is perpendicular to segment $BC$, and segment $DE$ is parallel to segment $AB$. Additionally, $\angle EAC = 60^\circ$.

![](https://cdn.mathpix.com/cropped/2024_05_01_4fd1b602786ad9981479g-37.jpg?height=637&width=622&top_left_y=1585&top_left_x=794)

a) Determine the value of $\angle EDC$.

b) Find the measure of angle $\angle AEC$.
Solution

a) Since $A E$ is tangent to the circle $c$, we have $\angle E A C = \angle A B C$ (angle of the segment and inscribed angle looking at the same arc). Given that $D E$ and $A B$ are parallel, the corresponding angles $\angle A B C$ and $\angle E D C$ are congruent, that is, $\angle E D C = 60^{\circ}$.

b) Consider the circle $d$ with diameter $A C$. Let $P$ be the intersection of the line $A E$ with the circle $d$. Since $\angle P A C$ and $\angle P D C$ are inscribed in the same arc, we must have

$$
\angle P A C = \angle P D C = 60^{\circ}
$$

From the previous item, $\angle P D C = \angle E D C$ and thus $P$ is also on the line supporting the segment $E D$. Therefore, $P$ is the intersection of the lines $A E$ and $E D$, that is, $P = E$. Finally, given that $A C$ is the diameter of $d$ and $E$ is on $d$, it follows that $\angle A E C = 90^{\circ}$.