In quadrilateral ABCD; 2∠A = 3∠B = 2∠C = 6∠D. Find all the angles of the quadrilateral.

It is given that in quadrilateral ABCD, 2∠A = 3∠B = 2∠C = 6∠D.

Let 2∠A = 3∠B = 2∠C = 6∠D = 6k (where 6k is the L.C.M. of 2, 3, and 6).

Now, 2∠A = 6k

∠A = $\dfrac{6k}{2}$

∠A = 3k ……………(1)

Similarly, 3∠B = 6k

∠B = $\dfrac{6k}{3}$

∠B = 2k ……………(2)

2∠C = 6k

∠C = $\dfrac{6k}{2}$

∠C = 3k ……………(3)

And, 6∠D = 6k

∠D = $\dfrac{6k}{6}$

∠D = k ……………(4)

As we know, the sum of all angles in a quadrilateral is 360°.

So,

⇒ ∠A + ∠B + ∠C + ∠D = 360°

Using equation (1), (2), (3) and (4),

⇒ 3k + 2k + 3k + k = 360°

⇒ 9k = 360°

⇒ k = $\dfrac{360°}{9}$

⇒ k = 40°

Thus:

∠A = 3k = 3 x 40° = 120°

∠B = 2k = 2 x 40° = 80°

∠C = 3k = 3 x 40° = 120°

∠D = k = 40°

Hence, the angles of the quadrilateral are ∠A = 120°, ∠B = 80°, ∠C = 120° and ∠D = 40°.