Question

In parallelogram LOST, SN ⊥ OL and SM ⊥ LT. Find ∠STM, ∠SON and ∠NSM.

Answer 1

Given, ∠MST = 40°

In ΔMST,

By the angle sum property of a triangle,

∠TMS + ∠MST + ∠STM = 180°

⇒ ∠STM = 180° – (90° + 40°)   ...[∵ SM ⊥ LT, ∠TMS = 90°]

= 50°

∴ ∠SON = ∠STM = 50°  ...[∵ Opposite angles of a parallelogram are equal]

Now, In the ΔONS,

∠ONS + ∠OSN + ∠SON = 180°  ...[Angle sum property of triangle]

∠OSN = 180° – (90° + 50°)

= 180° – 140°

= 40°

Moreover, ∠SON + ∠TSO = 180°   ...[∵ Adjacent angles of a parallelogram are supplementary]

⇒ ∠SON + ∠TSM + ∠NSM + ∠OSN = 180°

⇒ 50° + 40° + ∠NSM + 40° = 180°

⇒ 90° + 40° + ∠NSM = 180°

⇒ 130° + ∠NSM = 180°

⇒ ∠NSM = 180° – 130° = 50°