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

According to the properties of polygons, if a polygon has n sides, then the sum of its interior angles is (2n - 4) x 90°.

For a hexagon with 6 sides: n = 6

The sum of its interior angles is:

(2 x 6 - 4) x 90°

= (12 - 4) x 90°

= 8 x 90°

= 720° ……………(1)

It is given that hexagon ABCDEF has sides AB and EF parallel to each other.

As we know, corresponding angles are formed when a transversal crosses two lines, and the sum of corresponding angles is 180°. Therefore,

∠A + ∠F = 180° ……………(2)

It is also given that: ∠B : ∠C : ∠D : ∠E = 6 : 4 : 2 : 3.

Let the common factor of the angles be a. Thus, ∠B = 6a, ∠C = 4a, ∠D = 2a and ∠E = 3a.

Using the equations (1) and (2),

⇒ ∠A + ∠B + ∠C + ∠D + ∠E + ∠F = 720°

⇒ ∠A + 6a + 4a + 2a + 3a + ∠F = 720°

⇒ ∠A + ∠F + 15a = 720°

⇒ 180° + 15a = 720°

⇒ 15a = 720° - 180°

⇒ 15a = 540°

⇒ a = $\dfrac{540°}{15}$

⇒ a = 36°

Thus,

∠B = 6a = 6 x 36° = 216°

∠C = 4a = 4 x 36° = 144°

∠D = 2a = 2 x 36° = 72°

∠E = 3a = 3 x 36° = 108°

Hence, ∠B = 216°and ∠D = 72°.