question_answer 22)

In the below figure both RISK and CLUE are parallelograms. Find the value of \[x\].

Answer:

                \[\because \] RISK is a parallelogram \[\therefore \] \[\angle RIS=\angle RKS={{120}^{\text{o}}}\] |The opposite angles of a parallelogram are of equal measure                 Also,      \[\angle RIS=\angle ISK={{180}^{\text{o}}}\] |The adjacent angles in a parallelogram are supplementary \[\Rightarrow \]               \[{{120}^{\text{o}}}+\angle ISK={{180}^{\text{o}}}\] \[\Rightarrow \]               \[\angle IS={{180}^{\text{o}}}-{{120}^{\text{o}}}\] \[\Rightarrow \]               \[\angle ISK={{60}^{\text{o}}}\]                                ?(1) | The opposite angles of a parallelogram are of equal measure                                 In triangle EST, \[{{x}^{o}}+\angle TSE+\angle TES\,={{180}^{\text{o}}}\] |By angle sum property of a triangle \[\Rightarrow \]               \[{{x}^{\text{o}}}+\angle ISK+\angle CES={{180}^{\text{o}}}\] \[\Rightarrow \]               \[{{x}^{\text{o}}}+{{60}^{\text{o}}}+{{70}^{\text{o}}}={{180}^{\text{o}}}\]                 |From (1) and (2) \[\Rightarrow \]               \[{{x}^{\text{o}}}+{{130}^{\text{o}}}={{180}^{\text{o}}}\] \[\Rightarrow \]               \[{{x}^{\text{o}}}={{180}^{\text{o}}}-{{130}^{\text{o}}}={{50}^{\text{o}}}\Rightarrow \,x={{50}^{\text{o}}}\]