Let us analyze the diagram and solve the problem step by step.

Key Observations:

1. Given Information:

   • $ABCD$ is a parallelogram.
   • $ACED$ is a rhombus.
   • $\angle ABC = 50^\circ$.

2. Rhombus Properties:

   • In a rhombus, all sides are equal.
   • Diagonals of a rhombus bisect each other at right angles ($90^\circ$).
   • Diagonals of a rhombus bisect the angles at their vertices.

3. Parallelogram Properties:

   • Opposite angles are equal in a parallelogram.
   • Adjacent angles are supplementary.

Steps to Solve:

Step 1: Analyze $\triangle ACE$ in Rhombus $ACED$

• Since $ACED$ is a rhombus, diagonals bisect each other at $90^\circ$.
• $\angle AEC = 90^\circ$.

Step 2: Angle Relationships

• In parallelogram $ABCD$, adjacent angles are supplementary: $\angle ABC + \angle BCD = 180^\circ.$ Thus: $\angle BCD = 180^\circ - 50^\circ = 130^\circ.$

• Diagonal $AC$ bisects $\angle DCE$ (as $ACED$ is a rhombus): $\angle DCE = 130^\circ, \quad \text{so } \angle DCA = \frac{\angle DCE}{2} = \frac{130^\circ}{2} = 65^\circ.$

Step 3: Find $\angle X$

• In rhombus $ACED$, diagonals bisect vertex angles. So $\angle X = \angle DAE$, and: $\angle DAE = \frac{\angle DAB}{2}.$
• In parallelogram $ABCD$, opposite angles are equal: $\angle DAB = \angle BCD = 130^\circ.$ Thus: $\angle DAE = \frac{130^\circ}{2} = 65^\circ.$ Hence: $\angle X = 65^\circ.$

Final Answer:

$\boxed{\angle X = 65^\circ.}$

If you need further clarification or detailed verification, feel free to ask!

Expanding Questions:

1. How does the property of diagonals intersecting at $90^\circ$ in a rhombus simplify angle calculations?
2. What are the relationships between adjacent angles in a parallelogram?
3. Can you derive $\angle BCD$ using alternate interior angles instead?
4. Why do diagonals bisect the angles in a rhombus but not necessarily in a parallelogram?
5. How do supplementary angle relationships help in solving parallelogram problems?

Tip:

Always start by identifying given geometric properties (like parallelogram or rhombus) and apply their key characteristics to simplify the problem!