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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.

Work out the size of the angle marked x in the given geometry problem involving two parallel lines AB and CD, and a transversal EF. Given angles are 65° at angle AEF and 105° at angle FBD.

Solve the geometry problem involving parallel lines AB and CD, and a transversal EF. Find the angle x using corresponding angles theorem and linear pair axiom. Suitable for students in Grades 7-9.

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