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Prove that bisectors of any two adjacent angles of a parallelogram are at right angles.

Rectilinear Figures

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Answer

Let AC be bisector of ∠A and BD be bisector of ∠B and they meet at point M.

From figure,

Prove that bisectors of any two adjacent angles of a parallelogram are at right angles. Rectilinear Figures, ML Aggarwal Understanding Mathematics Solutions ICSE Class 9.

⇒ ∠A + ∠B = 180° (As AD || BC, sum of co-int ∠s = 180°)

A+B2=180°2=90.\dfrac{∠A + ∠B}{2} = \dfrac{180°}{2} = 90.

A2+B2=90°\dfrac{∠A}{2} + \dfrac{∠B}{2} = 90°

∴ ∠MAB + ∠MBA = 90° …….(i)

In △MAB,

⇒ ∠MAB + ∠MBA + ∠AMB = 180° (Sum of angles of triangle = 180°)

⇒ 90° + ∠AMB = 180° (from i)

⇒ ∠AMB = 180° - 90°

⇒ ∠AMB = 90°.

Hence, proved that bisectors of any two adjacent angles of a parallelogram are at right angles.

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