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) \[\Rightarrow \] \[y={{60}^{o}}\] \[\left| \text{Using (1)} \right.\] .
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