Question

pqrs is a parallelogram PS is produced to make M so that as M is equals to as r and M are produced to meet PQ PQ produce and prove that PQ and equals to QR​

Answer 1

Given that, PQRS is a parallelogram and line SR =SM

To Prove - Line QR = QN

Proof,

∠SRM = ∠RMS     [Since, angles opposite to equal sides of a triangle are equal]

∠QPS = ∠RQN = ∠SRM     [Co-interior angle]

∠SMR + ∠SRM = ∠PSR      [Exterior angle property]   -    (1)

∠SRQ + ∠PSR = 180°          [Since, sum of co-interior angles of a parallelogram is equal to 180°]

⇒ ∠SRQ + ∠SMR + ∠SRM = 180°    [From (1)]

⇒ ∠SRQ = 180° - ∠SMR - ∠SRM     -     (2)

∠QRM = ∠SRQ + ∠SRM      -     (3)

∠NRQ + ∠QRM = 180°        [Linear Pair]

⇒∠NRQ + ∠SRQ + ∠SRM = 180°     [From (3)]

⇒∠NRQ + 180° - ∠SMR - ∠SRM + ∠SRM = 180°    [From (2)]

⇒∠NRQ = ∠SMR

Since, two angels of both triangles are equal and  ∠SRM = ∠SMR,

∠NRQ = ∠SMR = ∠SRM = ∠RNQ    

Therefore, line QR = QN     [ Sides opposite to equal angles of a triangles are equal]