Dodecagon

Calculate the size of the smaller angles determined by lines A1 A4 and A2 A10 in the regular dodecagon A1 A2A3. .. A12. Express the result in degrees.

Final Answer:

x =  75 °

Step-by-step explanation:

$$\begin{gathered} n=12 \\ \phi =\frac{360}{n}=\frac{360}{12}=30{}^{\circ } \\ {A}_1=1\cdot \phi =1\cdot 30=30{}^{\circ } \\ {A}_2=2\cdot \phi =2\cdot 30=60{}^{\circ } \\ {A}_4=4\cdot \phi =4\cdot 30=120{}^{\circ } \\ {A}_{10}=10\cdot \phi =10\cdot 30=300{}^{\circ } \\ {s}_1=\frac{A_4-A_1}{2}=\frac{120-30}{2}=45{}^{\circ } \\ {s}_2=\frac{A_{10}-A_2}{2}=\frac{300-60}{2}=120{}^{\circ } \\ {x}_1=s_2-s_1=120-45=75{}^{\circ } \\ {x}_2=180-x_1=180-75=105{}^{\circ } \\ x=x_1=75=75^{\circ } \end{gathered}$$

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