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Let's solve the problem step by step.

### Given Information:
- \( ABCDEF \) is a hexagon.
- \( \text{Angle } BAF = \text{Angle } ABC = \text{Angle } AFE = \text{Angle } BCD \).
- \( \text{Angle } DEF = \text{Angle } CDE = 130^\circ \).

### Problem:
We are asked to find the size of angle \( BAF \).

### Approach:
1. **Step 1: Internal Angles of the Hexagon**:
   A regular hexagon has six interior angles. The sum of the interior angles of a polygon can be calculated by the formula:
   \[
   \text{Sum of interior angles} = (n - 2) \times 180^\circ,
   \]
   where \( n \) is the number of sides. For a hexagon, \( n = 6 \):
   \[
   \text{Sum of interior angles} = (6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ.
   \]
   So, the total sum of the angles in the hexagon is \( 720^\circ \).

2. **Step 2: Known Angles**:
   We are given that angle \( DEF = CDE = 130^\circ \). This accounts for two angles.

3. **Step 3: Remaining Angles**:
   Since we know two angles, the sum of the remaining four angles (which are equal) must add up to the remaining portion of the total angle sum. Thus:
   \[
   \text{Remaining sum} = 720^\circ - (130^\circ + 130^\circ) = 720^\circ - 260^\circ = 460^\circ.
   \]
   Since the four remaining angles are equal:
   \[
   \frac{460^\circ}{4} = 115^\circ.
   \]
   Therefore, the size of angle \( BAF \) is \( 115^\circ \).

### Final Answer:
The size of angle \( BAF \) is \( 115^\circ \).

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you calculate the interior angles of a polygon?
2. What is the sum of the interior angles of an octagon?
3. How would the problem change if the hexagon were irregular?
4. What is the relationship between exterior and interior angles in polygons?
5. Can you apply this method to find angles in any other polygons?

**Tip:** Always break down complex geometry problems by finding relationships between angles and sides first!

ABCD is a hexagon. Angle BAF = Angle ABC = Angle AFE = Angle BCD. Angle DEF = Angle CDE = 130°. Work out the size of angle BAF.

This page provides a step-by-step solution to finding the size of angle BAF in a regular hexagon where Angle DEF = Angle CDE = 130°. Learn about polygon angle sums, key geometric theorems, and angle relationships.

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