Question

The angle between the two altitudes of a parallelogram through the vertex of an obtuse angle of the parallelogram is 45°. Find the angles of the parallelogram.

Answer 1

Let ABCD be a parallelogram, where BE and BF are the perpendiculars through the vertex B to the sides DC and AD, respectively.

Let ∠A = ∠C = x, ∠B = ∠D = y  ...[Opposite angles are equal in parallelogram]

Now, ∠A + ∠B = 180°   ...[Adjacent sides of a parallelogram are supplementary]

In triangle ABF;

∠ABF = 90° – x

And in triangle BEC,

∠EBC = 90° – x

So, x + 90° – x + 45° + 90° – x = 180° 

⇒ – x = 180° – 225°

⇒ x = 45°

So, ∠A = ∠C = 45°

∠B = 45° + 45° + 45° = 135°

⇒ ∠D = 135°

Hence, the angles are 45°, 135°, 45° and 135°.