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.