The image provided appears to include a geometry problem involving a square $ABCD$ and an equilateral triangle $CDE$. Additionally, the handwritten solution shows calculations involving angles and relationships between the sides of the shapes. Here's an analysis and explanation of the problem based on the given diagram and notes:

Problem Details:

1. $ABCD$ is a square.
2. $\triangle CDE$ is an equilateral triangle (each angle is $60^\circ$).
3. The task is to find specific angles related to the given configuration.

Given Information and Steps:

1. Since $CDE$ is equilateral:

   • $\angle CDE = \angle DCE = \angle ECD = 60^\circ$.
   • All sides of $\triangle CDE$ are equal, i.e., $CE = DE = DC$.

2. In the square $ABCD$:

   • All sides are equal, i.e., $AB = BC = CD = DA$.
   • Each angle of the square is $90^\circ$.

3. Observing the angles in the diagram:

   • The key is to find angles involving vertices $A$, $B$, $C$, $D$, and $E$.

Solution in the Handwritten Notes:

1. Using properties of equilateral triangles and squares:

   • $\angle ADC = \angle DAB = 90^\circ$ (square property).
   • $\angle DCE = 60^\circ$ (equilateral triangle property).
   • $\angle ADC$ and $\angle DCE$ are adjacent at point $D$.

2. Angles and relationships derived:

   • $\angle ADE = 30^\circ$ (based on complementary and geometric properties).
   • $\angle AEB = 120^\circ$ (calculated in the notes as a supplementary angle).

3. Final answers:

   • $\angle ADE = 30^\circ$,
   • $\angle ABD = 90^\circ$,
   • $\angle AEB = 120^\circ$.

Do you need further clarifications?

Would you like a detailed step-by-step breakdown of each angle calculation, or an explanation of why the relationships hold?

Related Questions:

1. How do the properties of squares and equilateral triangles help in solving geometric problems?
2. What other relationships exist between the diagonals of a square and angles in the given configuration?
3. How is $\angle AEB$ calculated as $120^\circ$? Can it be verified in another way?
4. What happens if $\triangle CDE$ is not equilateral but isosceles? How does it affect the problem?
5. How can we extend this problem to three-dimensional geometry?

Tip:

When solving geometry problems involving multiple shapes, always start by identifying known properties (e.g., angles, side lengths) and relationships (e.g., parallel or perpendicular lines).