https://file.aiquickdraw.com/mathbot/file/1733248749857d5421dc4.jpg

To find the angle labeled "?", we will use the hint provided to analyze the geometry of the diagram step-by-step.

### Key Observations:
1. **Structure of Squares**:
   - Two squares are leaning against the rectangle. The top right corner of the left square is the critical point where we draw auxiliary lines.

2. **Auxiliary Lines**:
   - Draw a **vertical line** downward from the intersection point (top right corner of the left square) to the base of the square.
   - Draw a **horizontal line** leftward from the same intersection point.

3. **Right Triangles**:
   - These auxiliary lines create right triangles inside the structure, which will help to compute or determine relationships among the angles.

### Strategy:
1. By construction, the two squares have sides of equal length, so the geometry will involve right angles (90°) and complementary angles.
2. Use properties of rectangles, squares, and supplementary angles to deduce the value of the angle labeled "?".
3. Identify the relationships between the angles using the horizontal and vertical lines, and consider angle sums in triangles.

### Step-by-Step Solution:
1. Label the point of intersection at the top right corner of the left square as \( P \).
   - \( P \) is the vertex where the auxiliary lines meet.
   
2. The angle labeled "?" is adjacent to the angle formed between the slanted line and the horizontal auxiliary line.

3. The slanted line forms an angle with the vertical side of the square. Let this angle be \( \theta \).

4. In the right triangle formed by the horizontal auxiliary line, vertical auxiliary line, and the slanted line:
   - The angle between the slanted line and the horizontal auxiliary line is \( 90^\circ - \theta \).

5. The angle labeled "?" is complementary to \( \theta \), so:
   \[
   \text{Angle labeled "?"} = 90^\circ - \theta.
   \]

### Deduction:
If the slope of the slanted line (or more details about the exact geometry) is available, \( \theta \) can be determined explicitly. However, using the provided hint and geometry, the angle labeled "?" is dependent on the relative orientation of the slanted line.

Would you like me to solve further using trigonometric relationships or provide a general geometric explanation?

### Related Questions:
1. How can auxiliary lines help in solving geometry problems?
2. What is the role of complementary and supplementary angles in solving this?
3. How does the square’s symmetry simplify angle calculations?
4. Can the labeled angle "?" be calculated numerically with more information?
5. How are right triangles used to derive angles in geometric shapes?

### Tip:
When solving geometry problems with unknown angles, drawing auxiliary lines to form right triangles is often the best first step, as they reveal useful angle and length relationships.

Part (E) Two squares are stuck together and leaning against the side of the rectangle, find the angle labeled '?'. (Hint: Draw a vertical line down from the point of intersection at the top right corner of the left square, and draw a horizontal line left from the same point.)

Learn how to calculate the angle labeled '?' in a geometric arrangement involving two squares leaning against a rectangle. This problem covers concepts of right triangles, complementary angles, and auxiliary line construction. Suitable for Grades 8-10 students.

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