(i) ∠ABC = ∠Y = 100° [In a parallelogram opposite angles are equal]
∠x + ∠Y = 180° [In a parallelogram sum of the adjacent angles is equal to 180°]
∠x + 100° = 180°
∠x = 180°-100°
∠x = 80°
∠x = ∠z = 80° [In a parallelogram opposite angles are equal]
(ii) ∠PSR + ∠Y = 180° [In a parallelogram sum of the adjacent angles is equal to 180°]
∠Y + 50° = 180°
∠Y = 180°- 50°
∠Y = 130°
∠x = ∠Y = 130° [In a parallelogram opposite angles are equal]
∠PSR = ∠PQR = 50° [In a parallelogram opposite angles are equal]
∠PQR + ∠Z = 180° [Linear pair]
50° + ∠Z = 180°
∠Z = 180°-50°
∠Z = 130°
(iii) In ΔPMN
∠MPN + ∠PMN + ∠PNM = 180° [Sum of all the angles of a triangle is 180°]
30° + 90° + ∠z = 180°
∠z = 180°-120°
∠z = 60°
∠y = ∠z = 60° [In a parallelogram opposite angles are equal]
∠z = 180°- 120° [In a parallelogram sum of the adjacent angles is equal to 180°]
∠z = 60°
∠z + ∠NML = 180° [In a parallelogram sum of the adjacent angles is equal to 180°]
60° + 90°+ ∠x = 180°
∠x = 180°-150°
∠x = 30°
(iv) ∠x = 90° [vertically opposite angles are equal]
In ΔDOC
∠x + ∠y + 30° = 180° [Sum of all the angles of a triangle is 180°]
90° + 30° + ∠y = 180°
∠y = 180°-120°
∠y = 60°
∠y = ∠z = 60° [alternate interior angles are equal]
(v) ∠x + ∠POR = 180° [In a parallelogram sum of the adjacent angles is equal to 180°]
∠x + 80° = 180°
∠x = 180°- 80°
∠x = 100°
∠y = 80° [In a parallelogram opposite angles are equal]
∠QRS =∠x = 100°
∠QRS + ∠Z = 180° [Linear pair]
100° + ∠Z = 180°
∠Z = 180°-100°
∠Z = 80°
(vi) ∠y = 112° [In a parallelogram opposite angles are equal]
∠y + ∠TUV = 180° [In a parallelogram sum of the adjacent angles is equal to 180°]
∠z + 40° + 112° = 180°
∠z = 180°- 152°
∠z = 28°
∠z =∠x = 28° [alternate interior angles are equal]
