Let \(A_1 A_2 ... A_{16}\) be a regular 16-gon. Find angle \(A_7 A_9 A_{13}\).
Let \(A_1 A_2 \ldots A_{16}\) be a regular 16-gon.
Find angle \(A_7 A_9 A_{13}\).
\(\begin{array}{|lrcll|} \hline & 2\beta+ 4*\dfrac{360^\circ}{16} &=& 180^\circ \qquad (1) \\\\ & 2\alpha+ 2*\dfrac{360^\circ}{16} &=& 180^\circ \qquad (2) \\\\ \hline \\ (1)+(2): & 2\beta+ 4*\dfrac{360^\circ}{16} + 2\alpha+ 2*\dfrac{360^\circ}{16} &=& 180^\circ + 180^\circ \\\\ & 2\beta+ 2\alpha + 6*\dfrac{360^\circ}{16} &=& 360^\circ \\\\ & 2\beta+ 2\alpha + 135^\circ &=& 360^\circ \\\\ & 2\beta+ 2\alpha &=& 360^\circ - 135^\circ \\\\ & 2\beta+ 2\alpha &=& 225^\circ \\\\ & 2(\beta+ \alpha) &=& 225^\circ \\\\ & \beta+ \alpha &=& \dfrac{225^\circ}{2} \\\\ & \mathbf{ \beta+ \alpha} &=& \mathbf{112.5^\circ} \\\\ & \theta &=& \beta+ \alpha \\ & \mathbf{ \theta} &=& \mathbf{112.5^\circ} \\ \hline \end{array}\)
\(\angle A_7 A_9 A_{13} = \mathbf{112.5^\circ}\)
Let \(A_1 A_2 \ldots A_{16}\) be a regular 16-gon.
Find angle \(A_7 A_9 A_{13}\).
\(\begin{array}{|lrcll|} \hline & 2\beta+ 4*\dfrac{360^\circ}{16} &=& 180^\circ \qquad (1) \\\\ & 2\alpha+ 2*\dfrac{360^\circ}{16} &=& 180^\circ \qquad (2) \\\\ \hline \\ (1)+(2): & 2\beta+ 4*\dfrac{360^\circ}{16} + 2\alpha+ 2*\dfrac{360^\circ}{16} &=& 180^\circ + 180^\circ \\\\ & 2\beta+ 2\alpha + 6*\dfrac{360^\circ}{16} &=& 360^\circ \\\\ & 2\beta+ 2\alpha + 135^\circ &=& 360^\circ \\\\ & 2\beta+ 2\alpha &=& 360^\circ - 135^\circ \\\\ & 2\beta+ 2\alpha &=& 225^\circ \\\\ & 2(\beta+ \alpha) &=& 225^\circ \\\\ & \beta+ \alpha &=& \dfrac{225^\circ}{2} \\\\ & \mathbf{ \beta+ \alpha} &=& \mathbf{112.5^\circ} \\\\ & \theta &=& \beta+ \alpha \\ & \mathbf{ \theta} &=& \mathbf{112.5^\circ} \\ \hline \end{array}\)
\(\angle A_7 A_9 A_{13} = \mathbf{112.5^\circ}\)