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Question 178
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.

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Solution

Let ABCD be a parallelogram, where BE and BF are the perpendicular 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]
x+ABF+FBE+EBC=180
x+90x+45+90x=180
[InΔABF,ABF=90x and in ΔBEC,EBC=90x]
x=180225x=45A=C=45B=45+45+45=135D=135
Hence the angles are 45,135,45,135.

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