Q. 24.3( 12 Votes )

# The following figures are parallelograms. Find the degree values of the unknowns *x, y, z.*

Answer :

(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]

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