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If 2 : 3 is the ratio of two unequal angles of a parallelogram, then what will be the measures of all the angles of the parallelogram?

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72 °, 108 °, 108 °, 52 °

72 °, 108 °, 72 °, 108 °

72 °, 108 °, 72 °, 52 °

72 °, 58 °, 108 °, 52 °

answer is B.

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Detailed Solution

We have given a parallelogram ABCD such that the ratio of two unequal angles of the parallelogram is 2 : 3.
Since the ratio of two unequal angles of a parallelogram is 2 : 3, therefore we have,

∠ A = 2 x and ∠ B = 3 x

We know that opposite angles of a parallelogram are equal.

∴ ∠ A = ∠ C and ∠ B = ∠ D ⇒ ∠ C = 2 x and ∠ D = 3 x

Since a parallelogram is a quadrilateral, therefore using the angle sum property of a quadrilateral, we have,

∴ ∠ A + ∠ B + ∠ C + ∠ D = 360 ° ⇒ 2 x + 3 x + 2 x + 3 x + = 360 ° ⇒ 10 x = 360 ° ⇒ x = 36 °

∴ ∴ ∠ A = ∠ C = 2 × 36 ° =72 ° and ∠ B = ∠ D = 3 × 36 ° =108 °

Therefore, the measures of all the angles of the parallelogram are

72 °, 108 °, 72 °,

and

108 ° .

Hence, option (2) is correct.

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