Question

In the adjoining figure PQRS is a rhombus and RST is an equilateral triangle.T and Qlies on opposite sides of RS. If ∠SPQ= 72°, calculate the
value of ∠RQT and ∠SQT.
(FAST PLEASE)

Answer 1

Given : PQRS is a rhombus and RST is an equilateral triangle. T and Q lies on opposite sides of RS.  ∠SPQ= 72°,

To Find : value of ∠RQT and ∠SQT

Solution:

PQRS is a rhombus  PQ = QR = RS = PS

rhombus is a parallelogram also

∠SPQ = 72°  

=> ∠QRS = 72°   ( opposite angles are equal )

∠PQR = 108°   ( sum of adjacent angles = 180°)

∠QRS = 72°  , ∠SRT = 60°  ( Equilateral Triangle)

=> ∠QRT = 72 + 60 = 132°

QR = RT   ( ∵ QR = RS  and RS = RT )

=> ∠RQT = ∠RTQ

=> ∠RQT + ∠RTQ + ∠QRT = 180°

=> 2 ∠RQT + 132° = 180°

=>  2∠RQT  = 48°

=> ∠RQT  = 24°

∠QRS = 72°  

SR = RQ  => ∠RSQ = ∠RQS

∠RSQ + ∠RQS + ∠QRS = 180°

=> 2∠RQS + 72°    = 180°

=> ∠RQS =  54°

∠SQT = ∠RQS -  ∠RQT  

=> ∠SQT = 54°-   24°

=> ∠SQT = 30°

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