Given: ABCDEF is a hexagon.
To prove: ∠ A + ∠ B + ∠ C + ∠ D + ∠ E + ∠ F = 720°
Construction: Join AC, AD and AE.
Proof:
In Δ ABC, ∠ 1 + ∠ B + ∠ 5 = 180° [Sum of the three angles of a triangle is 180° ] …….(i)
In Δ ACD, ∠ 2 + ∠ 6 + ∠ 7 = 180° [Sum of the three angles of a triangle is 180° ] …..(ii)
In Δ ADE, ∠ 3 + ∠ 8 + ∠ 9 = 180° [Sum of the three angles of a triangle is 180° ] …….(iii)
In Δ AEF, ∠ 4 + ∠ 10 + ∠ F = 180° [Sum of the three angles of a triangle is 180° ]……..(iv)
Adding (i), (ii), (iii) and (iv), we get,
(∠ 1 + ∠ B + ∠ 5) + (∠ 2 + ∠ 6 + ∠ 7) + (∠ 3 + ∠ 8 + ∠ 9) + (∠ 4 + ∠ 10 + ∠ F)  = 720°
(∠ 1 + ∠ 2 + ∠ 3 + ∠ 4) + ∠ B + (∠ 5 + ∠ 6) + (∠ 7 + ∠ 8) + ∠ 9 + ∠ 10) + ∠ F = 720°
⇒ ∠ A + ∠ B + ∠ C + ∠ D + ∠ E + ∠ F = 720°