Answer:
(i)1. DE = 50 cm
2. Perimeter = 300 cm
(j) See below for proof.
Explanation:
Question (i)
Opposite sides of a parallelogram are equal in length.
⇒ AB = CD and AD = BC
If AB = 50 cm and AB = CD then CD = 50 cm.
If CD = DE then DE = 50 cm.
As AB = 50 cm and AB = AE then AE = 50 cm.
Perimeter = 2 AB + 2 (AE + ED)
= 2(50) + 2(50 + 50)
= 100 + 200
= 300 cm
Question (j)
Opposite angles in a parallelogram are equal.
⇒ S = Q = 114°
If TS bisects S then S₁ = 114° ÷ 2 = 57°
Adjacent angles in a parallelogram sum to 180°.
⇒ Q + P = 180°
⇒ 114° + P = 180°
⇒ P = 180° - 114°
⇒ P = 66°
If PT bisects P then P₂ = 66° ÷ 2 = 33°
Interior angles of a triangle sum to 180°.
⇒ T₁ + P₂ + S₁ = 180°
⇒ T₁ + 33° + 57° = 180°
⇒ T₁ + 90° = 180°
⇒ T₁ = 180° - 90°
⇒ T₁ = 90°
Hence proving that PT is perpendicular to ST.