2. Let $A_{1}, A_{2}, A_{3}, A_{4}, A_{5}$ and $A_{6}$ be six points on a circle in this order such that $\overline{A_{1} A_{2}}=\overline{A_{2} A_{3}}$, $\widetilde{A_{3} A_{4}}=\widehat{A_{4} A_{5}}$ and $\widehat{A_{5} A_{6}}=\widehat{A_{6} A_{1}}$, where $\widehat{A_{1} A_{2}}$ denotes the arc length of the arc $A_{1} A_{2}$ etc. It is also known that $\angle A_{1} A_{3} A_{5}=72^{\circ}$. Find the size of $\angle A_{4} A_{6} A_{2}$ in degrees.
2. Answer: 54 .
First, observe that $\angle A_{1} A_{6} A_{5}=180^{\circ}-72^{\circ}=108^{\circ}$. Hence $\angle A_{4} A_{6} A_{2}=\angle A_{1} A_{6} A_{5}-$ $\angle A_{1} A_{6} A_{2}-\angle A_{4} A_{6} A_{5}=108^{\circ}-\angle A_{2} A_{6} A_{3}-\angle A_{3} A_{6} A_{4}=108^{\circ}-\angle A_{4} A_{6} A_{2}$. Thus, $2 \angle A_{4} A_{6} A_{2}=108^{\circ}$, resulting in $\angle A_{4} A_{6} A_{2}=54^{\circ}$.