Dodecagon

Calculate the size of the smaller of the angles determined by lines A1 A4 and A2 A10 in the regular dodecagon A1A2A3. .. A12. Express the result in degrees.

Correct answer:

x =  75 °

Step-by-step explanation:

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

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