Let's break down the question:

You have a cyclic quadrilateral $ABCD$ inscribed in a circle. The tangent to the circle at the point $C$ is $CE$. The angles $\angle AEC = 50^\circ$ and $\angle ABD = 50^\circ$ are given, and we are asked to find the measure of $\angle ABC$.

Step-by-step solution:

1. Using Tangent-Secant Angle Theorem: The tangent at a point on a circle forms an angle with the chord drawn to that point, which is equal to the angle subtended by the chord on the opposite side of the circle.

   • From the question, the tangent at $C$ is $CE$.
   • By the Tangent-Secant Theorem, the angle $\angle AEC$ formed by the tangent and the chord $AC$ is equal to the angle subtended by the chord $AC$ at the opposite side, which is $\angle ABC$.

2. Applying Given Angles: We are given that $\angle AEC = 50^\circ$. Therefore, by the theorem mentioned above, we conclude that: $\angle ABC = 50^\circ$

Thus, the magnitude of $\angle ABC$ is $50^\circ$.

Would you like more details or any further clarifications on this problem? Here are some related questions to explore:

1. How can we use the properties of cyclic quadrilaterals to solve for other angles?
2. What is the significance of the Tangent-Secant Theorem in geometry?
3. How does the inscribed angle theorem apply to cyclic quadrilaterals?
4. What is the difference between a tangent and a secant to a circle in geometry?
5. How can the angle $\angle ABC$ change if the cyclic quadrilateral is not convex?

Tip: Always look for properties such as tangents, secants, and inscribed angles when solving geometry problems related to circles!