Question

The angle ∠ PDQ between the altitudes of a parallelogram, through the same vertex of an obtuse angle of the
parallelogram is 60°. Find all the angles of the parallelogram.

Answer 1

Let us consider a parallelogram, ABCD. Where, DP⊥AB and DQ ⊥BC.

Given ∠PDQ = 60°

In quadrilateral DPBQ

∠PDQ + ∠DPB + ∠B + ∠BQD = 360° [Sum of all the angles of a Quadrilateral is 360°]

60° + 90° + ∠B + 90° = 360°

∠B = 360° – 240°

∠B = 120°

∠B = ∠D = 120° [Opposite angles of parallelogram are equal]

∠B + ∠C = 180° [Sum of adjacent interior angles in a parallelogram is 180°]

120° + ∠C = 180°

∠C = 180° – 120° = 60°

∠A = ∠C = 70° (Opposite angles of parallelogram are equal)

∴ Angles of a parallelogram are 60°, 120°, 60°, 120°