To solve the problem of finding the angle $\angle CDA$ in the trapezium $ABCD$, let's break down the information provided and use trigonometry.

Given Information:

• $ABCD$ is a trapezium with the following dimensions:
  • $AB = 6.5 \, \text{cm}$ (vertical height from $A$ to $B$)
  • $BC = 9 \, \text{cm}$ (top side)
  • $CD = 6 \, \text{cm}$ (diagonal connecting $C$ to $D$)
  • $AD = 23 \, \text{cm}$ (bottom side)

Steps to Find $\angle CDA$:

1. Draw a Perpendicular Line:

   • Drop a perpendicular from point $C$ to line $AD$, and label the point of intersection as $E$.
   • This creates a right triangle $CED$ where $CE$ is the vertical height from $C$ to $AD$, and $ED$ is the horizontal distance between points $D$ and $E$.

2. Calculate $CE$:

   • Since $CE$ is the same height as $AB$, we have $CE = 6.5 \, \text{cm}$.

3. Calculate $ED$:

   • $ED$ is found by subtracting $BC$ from $AD$: $ED = AD - BC = 23 \, \text{cm} - 9 \, \text{cm} = 14 \, \text{cm}$

4. Use the Tangent Function to Find $\angle CDA$:

   • In $\triangle CED$, $\tan(\angle CDA) = \frac{CE}{ED}$: $\tan(\angle CDA) = \frac{6.5}{14}$ $\tan(\angle CDA) \approx 0.4643$

5. Calculate $\angle CDA$:

   • Find the inverse tangent (arctan) of 0.4643: $\angle CDA = \tan^{-1}(0.4643) \approx 25.0^\circ$
   • Therefore, $\angle CDA \approx 25.0^\circ$ (to 1 decimal place).

Final Answer:

$\boxed{25.0^\circ}$

Would you like further details or explanations on any part of this solution? Here are some additional related questions:

1. How can the cosine rule be applied to solve angle problems in trapeziums?
2. What other trigonometric identities are useful for solving angles in non-right triangles?
3. Can the sine rule be applied in this problem instead of tangent?
4. How does the Pythagorean theorem assist in solving problems involving right triangles in trapeziums?
5. What are other properties of trapeziums that might help in similar geometric problems?

Tip: Always label all points and sides when solving geometry problems, as it makes it easier to apply trigonometric functions correctly.