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Q.

The angle between the two altitudes of a parallelogram through the vertex of an obtuse angle of the parallelogram is 45°, as shown below.

Find the angles of the parallelogram.

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a

45 °, 135 °, 45 °, 135 ∘

b

65 °, 115 °, 65 °, 115 ∘

c

35 °, 145 °, 35 °, 145 ∘

d

55 °, 125 °, 55 °, 125 ∘

answer is A.

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Detailed Solution

Here, we are given a parallelogram ABCD, with its two altitudes BF and BE on the sides AD and DC respectively.
We have to determine all the interior angle measures.
To determine the angle measures, we will make use of the various properties related to angles in a polygon.
So, we know that the adjacent angle pairs of a parallelogram are supplementary.
Using this property, we have,

∠ A + ∠ B = 180 ° x + ∠ A B F + ∠ E B F + ∠ C B E = 180 ° ............. (1)

Next, in the

, using the angle sum property of a triangle, we get,

∠ A B F + ∠ B F A + ∠ B A F = 180 ° ∠ A B F + 90 ° + x = 180 ° ∠ A B F = 180 ° − 90 ° − x ∠ A B F = 90 ° − x ............. (2)

Again, applying the angle sum property of a triangle to the

△ B E C

, we get,

∠ C B E + ∠ B E C + ∠ B C E = 180 ° ∠ C B E + 90 ° + x = 180 ° ∠ C B E = 180 ° − 90 ° − x ∠ C B E = 90 ° − x ............. (3)

Substituting the values of

∠ A B F

and

∠ C B E

in the equation (1) and simplifying, we get,

x + 90 ° − x + 45 ° + 90 ° − x = 180 ° − x + 225 ∘ = 180 ∘ − x = 180 ° − 225 ° x = 45 °

From this value of x, we conclude that

∠ A = ∠ C = 45 °

.

∠ B = ∠ A B F + ∠ F B E + ∠ E B C ∠ B = 45 ° + 45 ° + 45 ° ∠ B = 135 °

Again, as we know that the opposite angle pairs are equal in a parallelogram, so we have,

∠ B = ∠ D = 135 °

. So, the all the angles of the parallelogram, ABCD are found to be

45 °, 135 °, 45 °

and

135 ∘

.
Hence, the correct option is (1).

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