The image provided contains a geometry problem with a diagram. The diagram shows two parallel lines $AB$ and $CD$, intersected by a transversal $EF$. Angles are marked on the diagram, specifically $65^\circ$ at angle $AEF$ and $105^\circ$ at angle $FBD$. The problem is asking to "Work out the size of the angle marked $x$".

Step-by-Step Solution

1. Identify Angles Using Parallel Lines and Transversals:

   • Since $AB$ is parallel to $CD$, and $EF$ is a transversal, corresponding angles and alternate interior angles will be equal.
   • $\angle AEF = 65^\circ$ (given).
   • $\angle EFB = 105^\circ$ (given).

2. Determine Angle $x$:

   • Angle $x$ is located between $EF$ and $CD$, adjacent to angle $65^\circ$ (marked as $AEF$).
   • Since $AB$ and $CD$ are parallel and $EF$ is a transversal, angle $x$ and $65^\circ$ are corresponding angles. Therefore, $x = 65^\circ$.

3. Double-Check Consistency:

   • Confirm the correctness by checking the sum of the angles at point $F$ which are on a straight line (linear pair) $\angle EFA + \angle AEF + \angle FBD = 180^\circ$.
   • $65^\circ + 105^\circ + x = 180^\circ$.

Answer:

The angle $x$ is $65^\circ$.

Would you like any further details or explanations on this solution?

Follow-up Questions:

1. What are alternate interior angles, and how are they related to parallel lines?
2. Can you explain why the sum of angles on a straight line equals $180^\circ$?
3. How would the problem change if $AB$ and $CD$ were not parallel?
4. What other angles can be calculated from this diagram using corresponding angles?
5. How can you use the angle sum property in a triangle to find unknown angles?

Tip:

Remember that when working with parallel lines, corresponding angles, alternate interior angles, and co-interior angles are key concepts that often help in solving geometry problems.