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in Quadrilaterals by (49.8k points)
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The angle between two altitudes of a parallelogram through the vertex of an obtuse angle of the parallelogram is 60º. Find the angles of the parallelogram.

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1 Answer

+1 vote
by (47.3k points)

According to the question,

ABCD is parallelogram,

DP ⊥ AB

DQ ⊥ BC.

∠PDQ = 60o

In quad. DPBQ,

Using angle sum property of a quadrilateral,

We have,

∠PDQ + ∠Q + ∠P + ∠B = 360o

60o + 90o + 90o + ∠B = 360o

240o + ∠B = 360o

∠B = 360o – 240o

∠B = 120o

Since, opposite angles in parallelogram are equal,

We have,

∠B = ∠D = 120o

Since, opposite sides are parallel in parallelogram,

We have,

AB||CD

Also, since sum of adjacent interior angles is 180o,

We have,

∠B + ∠C = 180o

120o + ∠C = 180o

∠C = 180o – 120o

∠C = 60o

Since, opposite angles in parallelogram are equal,

We have,

∠C = ∠A = 60o

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