Question

In the figure , if PQ ‖ RS , ∠MXQ = 135 and ∠MYR = 40 , find ∠XMY.

Answer 1

${\boxed{\large{\bold{Solution}}}}$

here we need to draw a line AB parallel to the line PQ through point M. Now AB || PQ , PQ || RS.

⟹Therefore , AB || RS

⠀⠀⠀⠀⠀Now , ∠QXM + ∠XMB = 180°

⠀⠀⠀⠀⠀(AB || PQ interior angles on the ⠀⠀⠀⠀⠀same side of the transversal XM)

⟹But , ⠀⠀⠀⠀⠀⠀∠QXM = 135°

⟹So , ⠀⠀⠀⠀⠀⠀⠀ ∠XMB = 45°(1)

⟹now ,⠀⠀⠀⠀⠀⠀ ∠BMY = ∠MYR

⠀⠀⠀⠀⠀⠀⠀ (AB || RS , alternate angles)

⟹therefore ,⠀⠀⠀⠀ ∠BMY = 40 ° (2)

Adding (1) and (2) ,you get

⠀⠀⠀⠀⠀∠XMB + ∠BMY = 45° + 40°

⟹That is , ⠀⠀⠀∠XMY = 85°