Question

abcdefghij is a regular decagon. it rotates clockwise about its center to form the polygon abcdefghij. match each angle of rotation to the point that a (the image of a) coincides with after that rotation. 252° 108° 135° 180° 288° 324°

Explanation:

Step1: Find central - angle of regular decagon

The central - angle of a regular decagon is $\frac{360^{\circ}}{10}=36^{\circ}$.

Step2: Analyze each rotation angle

For $252^{\circ}$:

$\frac{252^{\circ}}{36^{\circ}} = 7$. Rotating $252^{\circ}$ clock - wise, $A$ will coincide with $H$.

For $108^{\circ}$:

$\frac{108^{\circ}}{36^{\circ}} = 3$. Rotating $108^{\circ}$ clock - wise, $A$ will coincide with $D$.

For $135^{\circ}$:

Since $135^{\circ}\div36^{\circ}=\frac{135}{36}=\frac{15}{4}$, not a whole number, there is no exact point match.

For $180^{\circ}$:

$\frac{180^{\circ}}{36^{\circ}} = 5$. Rotating $180^{\circ}$ clock - wise, $A$ will coincide with $F$.

For $288^{\circ}$:

$\frac{288^{\circ}}{36^{\circ}} = 8$. Rotating $288^{\circ}$ clock - wise, $A$ will coincide with $I$.

For $324^{\circ}$:

$\frac{324^{\circ}}{36^{\circ}} = 9$. Rotating $324^{\circ}$ clock - wise, $A$ will coincide with $J$.

Answer:

$252^{\circ}$: $H$; $108^{\circ}$: $D$; $180^{\circ}$: $F$; $288^{\circ}$: $I$; $324^{\circ}$: $J$ (no match for $135^{\circ}$)