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.
Solution:
Given, the angle between two altitudes of a parallelogram through the vertex of an obtuse angle of the parallelogram is 60º
We have to find the angles of the parallelogram.
Consider a parallelogram ABCD
∠ADC and ∠ABC are the two obtuse angles of the parallelogram
DQ and DP are the two altitudes of the parallelogram
DP ⊥ AB
DQ ⊥ BC.
∠PDQ = 60
In quadrilateral DPBQ,
We know that according to the quadrilateral angle sum property, the sum of all the four interior angles is 360 degrees.
Sum of all interior angles of a quadrilateral is = 360º
We have,
∠PDQ + ∠Q + ∠P + ∠B = 360º
60 + 90 + 90 + ∠B = 360º
240 + ∠B = 360º
∠B = 360º - 240º
∠B = 120º
Since, opposite angles in parallelogram are equal,
∠B = ∠D = 120º
Since, opposite sides are parallel in parallelogram,
AB||CD
Also, since sum of adjacent interior angles is 180 degrees
∠B + ∠C = 180º
120 + ∠C = 180º
∠C = 180º - 120º
∠C = 60º
Since, opposite angles in parallelogram are equal,
∠C = ∠A = 60º
Therefore, areas of the parallelogram are 60º, 120º, 60º and 120º
✦ Try This: The angle between two altitudes of a parallelogram through the vertex of an obtuse angle of the parallelogram is 45º. Find the angles of the parallelogram.
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 8
NCERT Exemplar Class 9 Maths Exercise 8.3 Problem 3
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.
Summary:
The angle between two altitudes of a parallelogram through the vertex of an obtuse angle of the parallelogram is 60º. The angles of the parallelogram are 60º, 120º, 60º and 120º
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