What are the measures of angles x and y in the arrangement of the tiles?

Answer

To find the measures of angles x and y in the arrangement of the tiles, we first recognize that the outer tiles are regular 12-sided polygons (dodecagons). 1. **Calculating the Interior Angle of a Dodecagon:** The formula for the measure of an interior angle of a regular polygon is given by: \[ \text{Interior Angle} = \frac{(n - 2) \times 180}{n} \] where n is the number of sides. For a dodecagon (n=12): \[ \text{Interior Angle} = \frac{(12 - 2) \times 180}{12} = \frac{10 \times 180}{12} = 150^{\circ} \] 2. **Understanding the Angle Arrangement:** Since there are four outer dodecagons surrounding a common center, we look at how these angles interact around the center. Each corner where the dodecagons meet contributes to the angles x and y. 3. **Finding Angle x:** The angle x is one of the angles formed at the center where two dodecagons meet. The angle x can be calculated as follows: \[ x + x + 150 = 360 \] This accounts for the two angles from dodecagons and the 150-degree angle from the dodecagon itself. Thus: \[ 2x + 150 = 360 \] \[ 2x = 360 - 150 \] \[ 2x = 210 \] \[ x = 105^{\circ} \] 4. **Finding Angle y:** The angle y is at the intersection of the two dodecagons. Each dodecagon contributes its interior angle and the formation around this intersection point. Since it forms another vertex from the adjacent angles of the dodecagons, we calculate: \[ y + y + 150 + 150 = 360 \] \[ 2y + 300 = 360 \] \[ 2y = 360 - 300 \] \[ 2y = 60 \] \[ y = 30^{\circ} \] **Final Results:** \[ x = 105^{\circ} \text{ and } y = 30^{\circ} \]