Dodecagon

Calculate the size of the smaller angles determined by lines A1 A4 and A2 A10 in the regular dodecagon A1A2A3. .. 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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