In a hexagon, ABCDEF, side AB is parallel to side FE and `angleB : angleC : angleD : angleE=6:4 :2 : 3` Find `angleB and angleD`.

Answer

To solve the problem, we need to find the values of angle B and angle D in the hexagon ABCDEF, given that side AB is parallel to side FE and the angles B, C, D, and E are in the ratio 6:4:2:3.

### Step-by-Step Solution:

1. **Understanding the Properties of the Hexagon**:
Since AB is parallel to FE, we know that the sum of angles A and F will be 180 degrees. This is due to the property of alternate interior angles formed by a transversal line cutting through parallel lines.

**Hint**: Remember that when two lines are parallel, the angles on the same side of the transversal add up to 180 degrees.

2. **Sum of Interior Angles of a Hexagon**:
The formula to calculate the sum of the interior angles of a polygon is (n - 2) × 180 degrees, where n is the number of sides. For a hexagon (n = 6):
\[ \text{Sum of interior angles} = (6 - 2) \times 180 = 720 \text{ degrees} \]

**Hint**: Use the formula for the sum of interior angles to find the total angle measure for any polygon.

3. **Setting Up the Angles**:
Let the angles B, C, D, and E be represented in terms of a variable x based on their given ratio:
- Angle B = 6x
- Angle C = 4x
- Angle D = 2x
- Angle E = 3x

**Hint**: When dealing with ratios, express each part of the ratio in terms of a common variable.

4. **Expressing the Total Angle Sum**:
The total sum of the angles in the hexagon can be expressed as:
\[ \text{Angle A} + \text{Angle B} + \text{Angle C} + \text{Angle D} + \text{Angle E} + \text{Angle F} = 720 \text{ degrees} \]
Since angle A + angle F = 180 degrees (from step 1), we can substitute:
\[ 180 + (6x + 4x + 2x + 3x) = 720 \]

**Hint**: Substitute known values to simplify the equation.

5. **Simplifying the Equation**:
Combine the angles:
\[ 180 + 15x = 720 \]
Now, isolate x:
\[ 15x = 720 - 180 \]
\[ 15x = 540 \]
\[ x = \frac{540}{15} = 36 \text{ degrees} \]

**Hint**: Isolate the variable by performing inverse operations.

6. **Finding Angle B and Angle D**:
Now that we have x, we can find the angles:
- Angle B = 6x = 6 × 36 = 216 degrees
- Angle D = 2x = 2 × 36 = 72 degrees

**Hint**: Substitute the value of x back into the expressions for angles to find their measures.

### Final Answer:
- Angle B = 216 degrees
- Angle D = 72 degrees