In a hexagon abcdef, side AB is parallel to side FE and angle B:angle C:angle D:angle E=6:4:2:3 Find angle B and angle D

Given the problem, we have a hexagon \( ABCDEF \) where side \( AB \) is parallel to side \( FE \). The angles \( \angle B \), \( \angle C \), \( \angle D \), and \( \angle E \) are in the ratio \( 6:4:2:3 \).

### Step 1: Define the angles
Let the angles be \( \angle B = 6x \), \( \angle C = 4x \), \( \angle D = 2x \), and \( \angle E = 3x \).

### Step 2: Use the parallel lines condition
Since \( AB \parallel FE \), the corresponding angles must be equal. Therefore:
\[
\angle B = \angle E
\]
Thus:
\[
6x = 3x
\]

This equation is clearly contradictory unless \( x = 0 \), which is impossible in this context. Since the initial setup (that \( \angle B = \angle E \)) is contradictory with the given ratio, it seems that we might need to revisit our assumptions.

### Step 3: Use the sum of interior angles of the hexagon
The sum of the interior angles of a hexagon is given by:
\[
\text{Sum of interior angles} = (6 - 2) \times 180^\circ = 720^\circ
\]

Let's denote the other two angles \( \angle A \) and \( \angle F \) as \( \alpha \) and \( \beta \) respectively.

Now we can write:
\[
\alpha + 6x + 4x + 2x + 3x + \beta = 720^\circ
\]
\[
\alpha + \beta + 15x = 720^\circ
\]

### Step 4: Consider symmetry and parallel condition
Given that \( AB \parallel FE \), \( \alpha = \beta \).

So, let's assume:
\[
\alpha = \beta
\]
This means:
\[
2\alpha + 15x = 720^\circ
\]

Without further information about \( \alpha \) or any additional conditions, we cannot directly find \( x \) without making an assumption about \( \alpha \).

### Conclusion
Given the provided angles and the relationship:
- \( \angle B = 6x \)
- \( \angle D = 2x \)

We need more information to solve for the exact angles \( \angle B \) and \( \angle D \). However, if additional details about \( \alpha \) and \( \beta \) or another condition (like symmetry or other equal angles) are provided, we can proceed to solve the exact values of \( x \) and therefore \( \angle B \) and \( \angle D \).

Would you like to provide additional information, or should we assume a symmetric condition to proceed? 

### Relative Questions:
1. Can you confirm if there is any other relationship between the angles of the hexagon?
2. Are angles \( \alpha \) and \( \beta \) equal, or is there a different condition we should consider?
3. Would you like to explore the symmetric properties of hexagons in more detail?
4. Should we consider the exterior angles or any alternate parallel sides to gain more insight?
5. How would assuming \( \alpha \) and \( \beta \) as \( 120^\circ \) each affect the calculation?

### Tip:
When dealing with complex polygons like hexagons, leveraging symmetry or parallel line properties often simplifies the problem significantly.

Explore how to solve angle ratios in a hexagon ABCDEF where side AB is parallel to side FE. Understand the relationship between angles B, C, D, and E in the ratio 6:4:2:3. Includes detailed steps and mathematical analysis. Suitable for high school geometry students.

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