We are given the following:

• $AB \parallel DC$ and $AD \parallel BC$
• $\angle DCA = 65^\circ$
• $\angle CAB = 50^\circ$

We are tasked with finding the value of $x$, which likely represents an angle in the figure.

Step-by-Step Solution:

1. Identify known angles:

   • $\angle DCA = 65^\circ$
   • $\angle CAB = 50^\circ$

2. Use parallel line properties: Since $AB \parallel DC$ and $AD \parallel BC$, we can apply the properties of alternate interior and corresponding angles.

3. Consider triangle $\triangle ABC$: In $\triangle ABC$, the interior angles must sum to $180^\circ$. Therefore, we can find the unknown angle at point $C$, which is $\angle ACB$.

   $\angle ACB = 180^\circ - \angle CAB - \angle ABC$ Where $\angle ABC$ corresponds to $x$ (since it is formed between the parallel lines).

4. Calculate angle $ACB$: $\angle ACB = 180^\circ - 50^\circ - 65^\circ = 65^\circ$

Thus, the value of $x$ is: $x = 65^\circ$

Would you like further clarification on any step, or do you have any additional questions?

5 Related Questions:

1. What are alternate interior angles and how are they applied in this problem?
2. How do you calculate the sum of angles in a triangle?
3. What properties of parallel lines help in determining unknown angles?
4. How can you extend this method to solve for angles in quadrilaterals?
5. Can corresponding angles also be used in this type of geometry problem?

Tip: Always check the properties of parallel lines, as they provide relationships like alternate and corresponding angles that help in angle calculations.