Example 5
Show that the bisectors of angles of a parallelogram form a rectangle.
Given: ABCD is a parallelogram
AP, BP, CR, DR are bisectors of
∠ A , ∠ B, ∠ C, ∠ D respectively
To prove: PQRS is a rectangle
Proof: A rectangle is a parallelogram with one angle 90°
First we will prove PQRS is a parallelogram
Now,
AB ∥ DC
& AD is transversal
∴ ∠ A + ∠ D = 180°
Multiplying by half
1/2∠ A + 1/2∠ D = 1/2 × 180°
1/2∠ A + 1/2∠ D = 90°
∠ DAS + ∠ ADS = 90°
Now,
In Δ ADS
∠ DAS + ∠ ADS + ∠ DSA = 180°
90° + ∠ DSA = 180°
∠ DSA = 180° – 90°
∠ DSA = 90°
Also, lines AP & DR intersect
So,∠ PSR = ∠ DSA
∴ ∠ PSR = 90°
Similarly, we can prove that
∠ SPQ = 90° , ∠ PQR = 90° and ∠ SRQ = 90°.
So, ∠ PSR = ∠ PQR & ∠ SPQ = ∠ SRQ
∴ Both pair of opposite angles of PQRS are equal
So, PQRS is a parallelogram
Also, ∠ PSR = ∠ PQR = ∠ SPQ = ∠ SRQ = 90°.
∴ PQRS is a parallelogram in which one angle 90°
⇒ PQRS is a rectangle.
Hence proved

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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