question_answer 6)

                Find the angle measure \[x\] in the following figures.                                    (a)                 (b)                 (c)                 (d)

Answer:

                (a) \[x+{{50}^{o}}+{{130}^{o}}+{{120}^{o}}={{360}^{o}}\] |By angle sum property of a quadrilateral \[\Rightarrow \]               \[x+{{300}^{\text{o}}}={{360}^{\text{o}}}\] \[\Rightarrow \]               \[x={{360}^{\text{o}}}-{{300}^{\text{o}}}\] \[\Rightarrow \]               \[x={{60}^{\text{o}}}\]                 (b) \[x+({{180}^{\text{o}}}-{{90}^{\text{o}}})+{{60}^{\text{o}}}+{{70}^{\text{o}}}={{360}^{\text{o}}}\] |By linear pair property and angle sum property of a quadrilateral \[\Rightarrow \]               \[x+{{220}^{\text{o}}}={{360}^{\text{o}}}\] \[\Rightarrow \]               \[x={{360}^{\text{o}}}-{{220}^{\text{o}}}\] \[\Rightarrow \]               \[x={{140}^{\text{o}}}\] (c) \[x+{{30}^{\text{o}}}+x+({{180}^{\text{o}}}-{{30}^{\text{o}}})\] \[+({{180}^{\text{o}}}-{{60}^{\text{o}}})\,=(5-2)\,\times {{180}^{\text{o}}}\] |By linear pair property and angle sum property of a pentagon \[\Rightarrow \]               \[2x+{{30}^{\text{o}}}={{110}^{\text{o}}}+{{120}^{\text{o}}}={{540}^{\text{o}}}\] \[\Rightarrow \]               \[2x+{{260}^{\text{o}}}={{540}^{\text{o}}}\] \[\Rightarrow \]               \[2x={{540}^{o}}-{{260}^{o}}\] \[\Rightarrow \]               \[2x={{280}^{o}}\] \[\Rightarrow \]               \[x=\frac{{{280}^{\text{o}}}}{2}\] \[\Rightarrow \]               \[x={{140}^{\text{o}}}\] (d) \[x+x+x+x+x=(5-2)\times {{180}^{\text{o}}}\] |By angle sum property of a regular pentagon \[\Rightarrow \]               \[5x={{540}^{\text{o}}}\] \[\Rightarrow \]               \[x=\frac{{{540}^{\text{o}}}}{5}\] \[\Rightarrow \]               \[x={{108}^{\text{o}}}\]