a decagon and a dodecagon with equal side lengths have been placed together. what is the measure of angle y?

To find the measure of angle \( y \) formed by a decagon (10-sided polygon) and a dodecagon (12-sided polygon) placed together with equal side lengths, we can follow these steps:

1. **Interior Angle Formula**: The interior angle of a regular polygon can be found using the formula:
   \[
   \text{Interior angle} = \frac{(n - 2) \times 180^\circ}{n}
   \]
   where \( n \) is the number of sides of the polygon.

2. **Decagon's Interior Angle**: For a decagon (\( n = 10 \)):
   \[
   \text{Interior angle} = \frac{(10 - 2) \times 180^\circ}{10} = \frac{8 \times 180^\circ}{10} = 144^\circ
   \]

3. **Dodecagon's Interior Angle**: For a dodecagon (\( n = 12 \)):
   \[
   \text{Interior angle} = \frac{(12 - 2) \times 180^\circ}{12} = \frac{10 \times 180^\circ}{12} = 150^\circ
   \]

4. **External Angles**: Since \( y \) is likely formed by the external angles of the decagon and dodecagon, we find the external angles:
   \[
   \text{External angle} = 180^\circ - \text{Interior angle}
   \]
   - For the decagon: \( 180^\circ - 144^\circ = 36^\circ \)
   - For the dodecagon: \( 180^\circ - 150^\circ = 30^\circ \)

5. **Measure of Angle \( y \)**: The external angles of the decagon and dodecagon together form angle \( y \), so:
   \[
   y = 36^\circ + 30^\circ = 66^\circ
   \]

Thus, the measure of angle \( y \) is \( 66^\circ \).

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Would you like further details or clarification?

Here are 5 related questions:
1. What is the sum of the interior angles of a decagon?
2. How do you calculate the external angles of a regular polygon?
3. If the polygons had unequal side lengths, how would that affect angle \( y \)?
4. What is the measure of each angle in a regular hexagon?
5. Can you find the sum of exterior angles for any regular polygon?

**Tip**: The sum of exterior angles of any polygon is always \( 360^\circ \), regardless of the number of sides.

A decagon and a dodecagon with equal side lengths have been placed together. What is the measure of angle y?

Discover how to calculate the measure of angle y formed by a decagon and a dodecagon placed together with equal side lengths. This geometry problem involves finding interior and exterior angles using key formulas and is suitable for students in Grades 9-10.

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