Let's solve the problem step by step.

Given:

• ABCD is a convex quadrilateral.
• $\angle BAC = 50^\circ$
• $\angle CAD = 60^\circ$
• $\angle CBD = 30^\circ$
• $\angle BDC = 2S$

We are asked to find the angle $\angle AEB$.

Step 1: Determine $\angle ABC$ and $\angle BCD$

Using the sum of the angles in triangle $ABC$, we have: $\angle ABC = 180^\circ - \angle BAC - \angle CBD$ Substitute the given values: $\angle ABC = 180^\circ - 50^\circ - 30^\circ = 100^\circ$

Similarly, for triangle $ACD$, the sum of angles gives: $\angle ACD = 180^\circ - \angle CAD - \angle BDC$ Substitute the values, and let $\angle BDC = 2S$: $\angle ACD = 180^\circ - 60^\circ - 2S = 120^\circ - 2S$

Step 2: Use the given information about angles

Now, let's look at the point $E$, where the diagonals $AC$ and $BD$ intersect. By the properties of angles in a convex quadrilateral and intersecting diagonals, we have: $\angle AEB = 180^\circ - (\angle BAC + \angle CBD)$ Substituting the given angles: $\angle AEB = 180^\circ - (50^\circ + 30^\circ) = 100^\circ$

So, the measure of $\angle AEB$ is $100^\circ$.

Conclusion:

The angle $\angle AEB$ is $100^\circ$.

Would you like further details or have any questions about the solution?

Here are 5 related questions you can explore:

1. How does the intersection of diagonals affect the angles in a quadrilateral?
2. What is the significance of convex quadrilaterals in geometry?
3. Can you derive a formula for $\angle AEB$ if the angles of the quadrilateral are different?
4. How do internal and external angles of quadrilaterals relate to each other?
5. What are the special properties of cyclic quadrilaterals?

Tip: Always verify angle calculations using multiple methods, such as the angle sum property and triangle rules, to avoid errors in geometric problems.