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Let's break this problem down step by step to find the size of angle \( \angle ODC \).

### Key Information from the Problem:
- \( ABE \) and \( ADF \) are tangents to the circle.
- \( \angle DAB = 40^\circ \)
- \( \angle CBE = 75^\circ \)
- We are asked to find \( \angle ODC \).

### Step 1: Recognizing the Circle Theorems
- **Tangent-Secant Angle Theorem**: The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. Thus, \( \angle DAB \) is equal to \( \angle DCB \) (angle in the alternate segment).
  \[
  \angle DCB = 40^\circ
  \]

### Step 2: Use the Exterior Angle Theorem
- The exterior angle theorem tells us that the exterior angle of a triangle is equal to the sum of the opposite interior angles. From triangle \( BOC \):
  \[
  \angle CBE = 75^\circ = \angle BOC
  \]
  This tells us that \( \angle BOC = 75^\circ \).

### Step 3: Calculate \( \angle ODC \)
- In the cyclic quadrilateral \( OBDC \), opposite angles sum to \( 180^\circ \). Therefore:
  \[
  \angle BOC + \angle ODC = 180^\circ
  \]
  Substituting \( \angle BOC = 75^\circ \):
  \[
  75^\circ + \angle ODC = 180^\circ
  \]
  Solving for \( \angle ODC \):
  \[
  \angle ODC = 180^\circ - 75^\circ = 105^\circ
  \]

### Final Answer:
The size of \( \angle ODC \) is \( 105^\circ \).

---

Let me know if you would like further details or explanations on any step! Here are five related questions to deepen your understanding:

1. What is the alternate segment theorem and how is it used in geometry?
2. Can you apply the cyclic quadrilateral theorem to a different set of angles?
3. How do tangents to a circle relate to the radius of the circle at the point of contact?
4. What is the exterior angle theorem and how does it apply to other geometric shapes?
5. How would the problem change if the tangents \( ABE \) and \( ADF \) were not present?

**Tip**: In circle geometry, always look for alternate segments and cyclic quadrilaterals to simplify angle calculations.

Points B, C, and D lie on the circumference of a circle, center O. ABE and ADF are tangents to the circle. Given: Angle DAB = 40°, Angle CBE = 75°. Determine the size of angle ODC.

Learn how to calculate angle ODC in a circle geometry problem involving tangents and a cyclic quadrilateral. This problem covers key concepts such as the alternate segment theorem and the exterior angle theorem. Given angles DAB = 40° and CBE = 75°, find the missing angle ODC.

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