We are tasked with finding the size of angle $\angle ODC$. Let's go step by step using the information provided.

Given Information:

• $\angle DAB = 40^\circ$
• $\angle CBE = 75^\circ$
• $ABE$ and $ADF$ are tangents to the circle.
• $B, C, D$ are points on the circumference of the circle with center $O$.

Step 1: Analyze the Angles

• The tangents $ABE$ and $ADF$ imply that the angles formed between the tangent and the chord are related to the angles subtended by the same chord at the circumference.
• Specifically, we can use the tangent-chord theorem, which states that the angle between a tangent and a chord through the point of contact is equal to the angle subtended by the chord on the opposite side of the circle.

Step 2: Applying the Tangent-Chord Theorem

• By the tangent-chord theorem, $\angle DAB = \angle DCB$, so: $\angle DCB = 40^\circ$

Step 3: Analyze $\triangle BOC$

• $\triangle BOC$ is an isosceles triangle because $OB = OC$ (both are radii of the circle).
• In $\triangle BOC$, the angle subtended at $B$ is $75^\circ$, so the remaining two angles at $O$ and $C$ are equal.

Let the angle at $O$ be $\theta$. The sum of the interior angles of a triangle is $180^\circ$, so: $\theta + \theta + 75^\circ = 180^\circ$ This simplifies to: $2\theta = 105^\circ$ $\theta = 52.5^\circ$

Thus, $\angle BOC = 52.5^\circ$.

Step 4: Find $\angle ODC$

• $\angle ODC$ is the external angle of triangle $BOC$, so: $\angle ODC = \angle BOC = 52.5^\circ$

Thus, the size of $\angle ODC$ is 52.5 degrees.

Do you want further details or have any questions about this solution?

Here are 5 related questions:

1. How does the tangent-chord theorem apply to different circle configurations?
2. What are the properties of an isosceles triangle, and how do they apply here?
3. How can you calculate the size of angles subtended by arcs in a circle?
4. What role do external angles play in solving circle geometry problems?
5. How does the sum of angles in a triangle help in solving circle-related questions?

Tip: Always remember the properties of tangents and circles—especially the relationship between tangents and angles subtended by chords, as they are essential for many circle geometry problems.