In quadrilateral ABCD, sides AB and DC are parallel to each other. If ∠A : ∠D = 2 : 3 and ∠B : ∠C = 7 : 8; find all the angles of quadrilateral ABCD.

It is given that in quadrilateral ABCD, the ratio of ∠A to ∠D is 2 : 3.

Let ∠A and ∠D be 2x and 3x, respectively.

Since, AB is parallel to DC, the sum of the interior angles on the same side of the transversal line is 180°.

Thus:

∠A + ∠D = 180°

⇒ 2x + 3x = 180°

⇒ 5x = 180°

⇒ x = $\dfrac{180°}{5}$

⇒ x = 36°

∠A = 2x = 2 x 36° = 72°

∠D = 3x = 3 x 36° = 108°

It is also given that the ratio of ∠B to ∠C is 7 : 8.

Let ∠B and ∠C be 7y and 8y, respectively.

Since AB is parallel to DC, the sum of the interior angles on the same side of the transversal line is 180°.

Thus:

∠B + ∠C = 180°

⇒ 7y + 8y = 180°

⇒ 15y = 180°

⇒ y = $\dfrac{180°}{15}$

⇒ y = 12°

∠B = 7y = 7 x 12° = 84°

∠D = 8y = 8 x 12° = 96°

Hence, the angles of the quadrilateral are ∠A = 72°, ∠B = 84°, ∠C = 96° and ∠D = 108°.