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The problem involves regular pentagonal tiles and triangular tiles arranged in a pattern. The goal is to find the angle measures of the triangular tiles and explain the reasoning.

### **Solution:**

1. **Understanding the Geometry:**
   - A regular pentagon has five equal sides and five equal interior angles.
   - The interior angle of a regular pentagon can be calculated using the formula:
     \[
     \text{Interior Angle} = \frac{(n-2) \times 180^\circ}{n}
     \]
     where \( n \) is the number of sides.
   - For a pentagon (\( n = 5 \)):
     \[
     \text{Interior Angle of Pentagon} = \frac{(5-2) \times 180^\circ}{5} = \frac{3 \times 180^\circ}{5} = 108^\circ
     \]

2. **Analyzing the Pattern:**
   - The angles \( D \) and \( A \) marked on the triangular tiles are formed where the sides of the pentagons meet the triangles.
   - Since each vertex of the triangular tile is shared by one pentagon, the sum of the angles around each vertex should be \( 360^\circ \).

3. **Determining the Triangular Tile Angles:**
   - Let's denote the angle of the triangular tile as \( x \). 
   - At each vertex of the triangular tile, there are two adjacent pentagons. The sum of the angles around the vertex is:
     \[
     x + 108^\circ + 108^\circ = 360^\circ
     \]
   - Simplifying this:
     \[
     x + 216^\circ = 360^\circ
     \]
     \[
     x = 360^\circ - 216^\circ = 144^\circ
     \]
   - Thus, each angle in the triangular tiles measures \( 144^\circ \).

### **Conclusion:**
The angles of the triangular tiles in this pattern are all \( 144^\circ \). The reasoning is based on the fact that the interior angles of the pentagons are \( 108^\circ \) each, and the sum of angles around a vertex is \( 360^\circ \).

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Would you like further details on this solution? Here are some additional questions related to this topic:

1. What is the general formula for finding the interior angle of any regular polygon?
2. How do the angles in a regular hexagon compare to those in a regular pentagon?
3. Can you calculate the exterior angle of a regular pentagon?
4. How would the angle calculations change if the pattern used hexagons instead of pentagons?
5. What is the sum of the interior angles in a regular polygon with 10 sides?

**Tip:** When working with regular polygons, remember that all interior angles are equal, and this symmetry often simplifies solving problems involving patterns and tiling.

Explore the geometric pattern involving regular pentagonal and triangular tiles. Learn how to calculate the angles of triangular tiles based on the interior angles of pentagons and vertex angle properties. Suitable for high school students studying geometry and patterns in regular polygons.

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