The problem involves finding the measure of angle $\angle AEC$ in the given geometric figure. The figure shows a circle with points $A$, $B$, $C$, and $E$ on its circumference, and $AB$ is a chord of the circle.

Solution:

1. Identify the type of angle $\angle AEC$:

   • $\angle AEC$ is an inscribed angle in the circle.

2. Relation between inscribed angle and arc:

   • The inscribed angle $\angle AEC$ intercepts the arc $AC$.
   • The measure of an inscribed angle is half the measure of the intercepted arc.

3. Given information:

   • In the figure, it seems that $E$ is placed such that $\angle AEB$ might be a straight line, making $E$ the point of intersection of the extensions of $AB$ and $CD$. If $AB$ is a diameter, then $\angle ACB = 90^\circ$, and we might use properties of cyclic quadrilaterals or specific arcs if more information were given.

4. Finding $\angle AEC$:

   • If $AB$ is the diameter, then arc $AC$ is 180°, meaning $\angle AEC = \frac{1}{2} \times 180^\circ = 90^\circ$.
   • Therefore, $\angle AEC$ is $90^\circ$.

The measure of $\angle AEC$ is $90^\circ$.

Would you like more details or have any questions?

Related Questions:

1. How can the properties of a cyclic quadrilateral be used to solve problems involving circles?
2. What is the relationship between an inscribed angle and the arc it intercepts?
3. How do we find angles in a circle when the chord is a diameter?
4. What is the significance of a diameter in a circle related to angles and arcs?
5. How does the concept of supplementary angles apply to cyclic quadrilaterals?

Tip:

Remember that an inscribed angle is always half the measure of the intercepted arc. This is a fundamental property when working with problems involving circles.