The diagram below shows four congruent squares and some of their diagonals. Let $T$ be the number of triangles and $R$ be the number of rectangles that appear in the diagram. Find $T + R$.
[img]https://cdn.artofproblemsolving.com/attachments/1/5/f756bbe67c09c19e811011cb6b18d0ff44be8b.png[/img]
1. **Counting the Triangles:**
   - Each square is divided into 4 triangles by its diagonals.
   - There are 4 squares, so the total number of triangles formed by the diagonals within the squares is \(4 \times 4 = 16\).
   - However, some triangles are counted multiple times. Specifically, the triangles formed by the intersections of the diagonals of adjacent squares are counted twice.
   - Each pair of adjacent squares shares 2 triangles at their intersection. There are 4 pairs of adjacent squares, so we subtract \(4 \times 2 = 8\) triangles.
   - Therefore, the total number of unique triangles is \(16 - 8 = 8\).

2. **Counting the Rectangles:**
   - Each square itself is a rectangle, so there are 4 rectangles from the squares.
   - Each pair of adjacent squares forms a larger rectangle. There are 4 such pairs, so we add 4 more rectangles.
   - The entire figure, which is composed of 4 squares, forms one large rectangle.
   - Additionally, each pair of diagonals within a square forms 2 smaller rectangles. Since there are 4 squares, this adds \(4 \times 2 = 8\) rectangles.
   - Therefore, the total number of rectangles is \(4 + 4 + 1 + 8 = 17\).

3. **Summing the Triangles and Rectangles:**
   - The total number of triangles \(T\) is 8.
   - The total number of rectangles \(R\) is 17.
   - Therefore, \(T + R = 8 + 17 = 25\).

The final answer is \(\boxed{25}\).