question_answer 7)

Find the values of the unknowns \[x\] and \[y\] in the following diagrams: (i)                (ii)                       (iii) (iv)           (v)                  (vi)                

Answer:

                (i)            \[{{50}^{o}}+x={{120}^{o}}\] \[\left| \text{By exterior}-\text{angle property of a triangle }...(1) \right.\] \[\Rightarrow \]               \[x={{120}^{o}}-{{50}^{o}}\] \[\Rightarrow \]               \[x={{70}^{o}}\]                                                                ? (2) Again,   \[x+y+{{50}^{o}}={{180}^{o}}\] \[\Rightarrow \]               \[x+y+{{50}^{o}}={{180}^{o}}\] \[\left| \text{By angle}-\text{sum property of a triangle }...(4) \right.\] \[\Rightarrow \]               \[x+y={{180}^{o}}-{{50}^{o}}\] \[\Rightarrow \]               \[x+y={{130}^{o}}\]                        \[\Rightarrow \]               \[\text{7}{{\text{0}}^{o}}+y={{130}^{o}}\]                           \[\left| \text{Using (2)} \right.\] \[\Rightarrow \]               \[y={{130}^{o}}-{{70}^{o}}\] \[\Rightarrow \]               \[y={{60}^{o}}\]                                                                ? (5) (ii)           \[y={{80}^{o}}\] \[\left| \text{Vertically opposite angles are equal }...(1) \right.\] \[\Rightarrow \]               \[x+{{50}^{o}}+y={{180}^{o}}\] \[\left| \text{By angle}-\text{sum property of a triangle} \right.\] \[\Rightarrow \]               \[x+y={{180}^{o}}-{{50}^{o}}\] \[\Rightarrow \]               \[x+y={{130}^{o}}\] \[\Rightarrow \]               \[x+{{80}^{o}}={{130}^{o}}\]                       \[\left| \text{Using(1)} \right.\] \[\Rightarrow \]               \[x={{130}^{o}}-{{80}^{o}}\] \[\Rightarrow \]               \[x={{50}^{o}}\] (iii)          \[x={{50}^{o}}+{{60}^{o}}\]         \[\left| \text{By exterior-angle property of atriangle} \right.\] \[\Rightarrow \]               \[x={{110}^{o}}\] \[\Rightarrow \]               \[x=110\] \[y+{{50}^{o}}+{{60}^{o}}={{180}^{o}}\]                                \[\left| \text{By angle-sum property of a triangle} \right.\]                 \[\Rightarrow \]               \[y+{{110}^{o}}={{180}^{o}}\]                 \[\Rightarrow \]               \[y={{180}^{o}}-{{110}^{o}}\]                 \[\Rightarrow \]               \[y={{70}^{o}}\]                 (iv)         \[x={{60}^{o}}\]                                                ?.(1)                                 \[\left| \text{Vertically opposite angles are equal} \right.\]                         \[x+{{30}^{o}}+y={{180}^{o}}\]  \[\left| \text{By angles-sum property of triangle} \right.\]                                 \[x+y={{180}^{o}}-{{30}^{o}}\]                 \[\Rightarrow \]               \[x+y={{150}^{o}}\]                 \[\Rightarrow \]               \[{{60}^{o}}+y={{150}^{o}}\]                       \[\left| \text{Using (1)} \right.\]                 \[\Rightarrow \]               \[y={{150}^{o}}-{{60}^{o}}\]                 \[\Rightarrow \]               \[y={{90}^{o}}\]                 (v)          \[y={{90}^{o}}\]                                                ? (1) \[\left| \text{Vertically opposstie angles are equal} \right.\] \[x+x+y={{180}^{o}}\]   \[\left| \text{By angle-sum property of a triangle} \right.\] \[\Rightarrow \]               \[2x+y={{180}^{o}}\] \[\Rightarrow \]               \[2x+{{90}^{o}}={{180}^{o}}\]                    \[\left| \text{Using (1)} \right.\] \[\Rightarrow \]               \[2x={{180}^{o}}-{{90}^{o}}\] \[\Rightarrow \]               \[2x={{90}^{o}}\] \[\Rightarrow \]               \[x=\frac{{{90}^{o}}}{2}\] \[\Rightarrow \]               \[x={{45}^{o}}\] (vi)         \[x=y\]                                                 ? (1) \[x+x+y={{180}^{o}}\]   \[\left| \text{Vertically oppostie angles are equal} \right.\] \[\Rightarrow \]               \[2x+y={{180}^{o}}\]      \[\left| \text{By angle-sum property of a triangle} \right.\]                 \[\Rightarrow \]               \[2x+x={{180}^{o}}\]                      \[\left| \text{Using (1)} \right.\]                 \[\Rightarrow \]               \[3x={{180}^{o}}\] \[\Rightarrow \]               \[3x=\frac{{{180}^{o}}}{3}\] \[\Rightarrow \]               \[x={{60}^{o}}\]                                                                ? (2) \[\Rightarrow \]               \[y={{60}^{o}}\]                                                \[\left| \text{Using (1)} \right.\] .