In the given problem, we are asked to calculate the size of angle $\theta$, with $CDE$ as a tangent to the circle at point $D$.

Key Information:

• $CDE$ is a tangent, and we are to use properties related to circles and tangents.
• $\angle BDA = 56^\circ$ (this is an exterior angle formed between the tangent and a chord).
• $\angle DBA = 37^\circ$ is given.
• $\angle DAF = 29^\circ$, part of the cyclic quadrilateral inside the circle.

Step-by-Step Solution:

1. Using the Tangent-Secant Theorem:

   • The angle between the tangent and a chord (in this case, $\angle BDA$) is equal to the angle in the alternate segment of the circle. Therefore, $\angle BDA = \angle B$.
   • Since $\angle BDA = 56^\circ$, we can conclude that $\angle B = 56^\circ$.

2. Angles in a Triangle:

   • Now, consider triangle $ABD$ inside the circle. The angles in this triangle must sum to $180^\circ$. $\angle A + \angle B + \angle DBA = 180^\circ$ Substituting the known values: $\theta + 56^\circ + 37^\circ = 180^\circ$ Simplifying this equation: $\theta + 93^\circ = 180^\circ$ $\theta = 180^\circ - 93^\circ = 87^\circ$

Conclusion:

The size of angle $\theta$ is $87^\circ$.

Let me know if you'd like any further details or explanations!

Here are some follow-up questions to reinforce the concept:

1. How does the tangent-secant theorem apply to different types of circles?
2. Why do the angles in a triangle always sum to 180 degrees?
3. What are some other properties of cyclic quadrilaterals?
4. How would the solution change if the tangent was extended beyond point $E$?
5. Can you apply this method to solve for angles in an ellipse?

Tip: Always remember that the angle between a tangent and a chord is equal to the angle in the alternate segment of the circle. This property is key to solving many circle geometry problems!