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In the adjoining figure, ABCD is a rhombus whose diagonals intersect at O. If ∠OAB : ∠OBA = 2 : 3, find the angles of △ OAB.

In the adjoining figure, ABCD is a rhombus whose diagonals intersect at O. If ∠OAB : ∠OBA = 2 : 3, find the angles of △ OAB. Quadrilaterals, R.S. Aggarwal Mathematics Solutions ICSE Class 9.

Rectilinear Figures

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Answer

Let ∠OAB = 2x and ∠OBA = 3x.

The diagonals of rhombus are perpendicular to each other.

∴ ∠AOB = 90°

In △AOB,

⇒ ∠AOB + ∠OAB + ∠OBA = 180°

⇒ 90° + 2x + 3x = 180°

⇒ 2x + 3x = 180° - 90°

⇒ 5x = 90°

⇒ x = 90°5\dfrac{90°}{5}

⇒ x = 18°.

∠OAB = 2x = 2(18°) = 36°.

∠OBA = 3x = 3(18°) = 54°.

Hence, ∠OAB = 36°, ∠OBA = 54°, ∠AOB = 90°.

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