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The angle between the altitudes of a parallelogram, through the same vertex of an obtuse angle of the parallelogram is $60 ° .$ Find the angles of the parallelogram.

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°, 130 °, 50 °, 130 °

60 °, 120 °, 60 °, 120 °

°, 140 °, 40 °, 147 °

°, 150 °, 30 °, 150 °

answer is B.

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Detailed Solution

Given that the angle between the altitudes of a parallelogram, through the same vertex of an obtuse angle of the parallelogram is

60 ° .

Let ABCD is the parallelogram with altitudes DQ and DP.

∴ ∠ PDQ = 60 °

By applying angle sum property in quadrilateral D P B Q, we get, ∠ P D Q + ∠ D P B + ∠ B + ∠ B Q D = 360 ° ⇒ 60 ° + 90 ° + ∠ B + 90 ° = 360 ° ⇒ ∠ B + 240 ° = 360 ° ⇒ ∠ B = 360 ° − 240 ° ⇒ ∠ B = 120 °

Also, we know that the sum of adjacent angles in a parallelogram is 180

° .

⇒ ∠ B + ∠ C = 180 ° ⇒ 120 ° + ∠ C = 180 ° ⇒ ∠ C = 180 ° − 120 ° ⇒ ∠ C = 60 ° As we know that the opposite angles of a parallelogram are equal . ∴ ∠ A = ∠ C = 60 ° . Similarly, ∠ B = ∠ D = 120 ° .

The angles of parallelogram are

60 °, 120 °, 60 °, 120 °

.
Hence, the correct option is 2.

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