In a figure, PQRS is a parallelogram and angle SPQ=60 degreee If the bisectors of angle P and angle - Maths - Quadrilaterals

Question

In a figure, PQRS is a parallelogram and angle SPQ=60 degreee If
the bisectors of angle P and angle Q meet at A on RS, prove that A
is the midpoint of RS

Poonam Mittal · Accepted Answer

We have âˆ  P = 60Â°
and âˆ  P + âˆ  Q =
180Â° â‡’âˆ  Q =
120Â° (sum of interior angles on the same side of
transversal)
AP is the angle bisector
âˆ´ âˆ SPA =
âˆ APQ = 30Â° (a)
similarly âˆ AQR = âˆ AQP =
60Â°
PQ is parallel to SR, and PA is a transversal
âˆ´âˆ SAP =
âˆ APQ = 30Â° (alternate angles)
âˆ´âˆ SAP =
âˆ SPA (from a)
Now in Î”PAS,
âˆ SAP = âˆ SPA
âˆ´ SA = PS (by isosceles triangle property)
(1)
Similarly AR=RQ (2)
also PS = RQ (sides of parallelogram) (3)
by (1), (2), and (3)
SA = AR
âˆ´ A is the mid point of SR

In a figure, PQRS is a parallelogram and angle SPQ=60 degreee. If the bisectors of angle P and angle Q meet at A on RS, prove that A is the midpoint of RS...