Question

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

Answer 1

Given ABCD is a parallelogram in which DP⊥AB and AQ ⊥BC.

Given ∠PDQ = 60°

In quad. 

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

60° + 90° + ∠B + 90° = 360°

∠B = 360° – 240°

Therefore, ∠B = 120°

But ∠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°

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