Bisectors of angles of a parallelogram form a rectangle
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Show that the bisectors of angles of a parallelogram form a rectangle
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Given : A parallelogram ABCD.
To prove : PQRS is a rectangle.
Proof :
In ΔABS ,
1/2∠A + 1/2∠B + ∠BSA = 180 °
∠A+∠B = 180 ° (Adjacent angles of a parallelogram are supplementary )
∴ 1/2∠A + 1/2∠B = 180 ÷ 2 = 90 °
90 ° + ∠BSA = 180 °
∴ ∠BSA = 180° - 90°
∠BSA = 90°
∠BSA = ∠RSP [Vertically opposite angles ]
∠RSP = 90 °
Similarly , it can be showed that ∠SPQ = 90° , ∠PQR = 90° and ∠QRS = 90°
∴ PQRS is a quadrilateral in which all the angles are of measure 90°.
Hence , PQRS is a rectangle
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