(i) ∠ABC = ∠y = 100^o (opposite angles are equal in a parallelogram)

∠x + ∠y = 180^o (sum of adjacent angles is = 180^o in a parallelogram)

∠x + 100^o = 180^o

∠x = 180^o – 100^o

= 80^o

∴ ∠x = 80^o ∠y = 100^o∠z = 80^o (opposite angles are equal in a parallelogram)

(ii) ∠RSP + ∠y = 180^o (sum of adjacent angles is = 180^o in a parallelogram)

∠y + 50^o = 180^o

∠y = 180^o – 50^o

= 130^o

∴ ∠x = ∠y = 130^o (opposite angles are equal in a parallelogram)

∠RSP = ∠RQP = 50^o (opposite angles are equal in a parallelogram)

∠RQP + ∠z = 180^o (linear pair)

50^o + ∠z = 180^o

∠z = 180^o – 50^o

= 130^o

∴ ∠x = 130^o ∠y = 130^o ∠z = 130^o

(iii) In ΔPMN

∠NPM + ∠NMP + ∠MNP = 180° [Sum of all the angles of a triangle is 180°]

30° + 90° + ∠z = 180°

∠z = 180°-120°

= 60°

∠y = ∠z = 60° [opposite angles are equal in a parallelogram]

∠z = 180°-120° [sum of the adjacent angles is equal to 180° in a parallelogram]

∠z = 60°

∠z + ∠LMN = 180° [sum of the adjacent angles is equal to 180° in a parallelogram]

60° + 90°+ ∠x = 180°

∠x = 180°-150°

∠x = 30°

∴ ∠x = 30^o ∠y = 60^o ∠z = 60^o

(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]

∴ ∠x = 90^o ∠y = 60^o ∠z = 60^o

(v) ∠x + ∠POR = 180° [sum of the adjacent angles is equal to 180° in a parallelogram]

∠x + 80° = 180°

∠x = 180°-80°

∠x = 100°

∠y = 80° [opposite angles are equal in a parallelogram]

∠SRQ =∠x = 100°

∠SRQ + ∠z = 180° [Linear pair]

100° + ∠z = 180°

∠z = 180°-100°

∠z = 80°

∴ ∠x = 100^o ∠y = 80^o ∠z = 80^o

(vi) ∠y = 112° [In a parallelogram opposite angles are equal]

∠y + ∠VUT = 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]

∴ ∠x = 28^o ∠y = 112^o ∠z = 28^o