Let's analyze the problem step-by-step based on the provided image.

Given Information:

1. $ABCD$ is a cyclic quadrilateral.
2. Angle $\angle BCD = 100^\circ$.
3. $EF$ is a tangent to the circle, touching it at point $A$.
4. Angle $\angle DAF = 30^\circ$.

We are asked to find:

• a) $\angle BAD$
• b) $\angle EAB$

Solution:

1. Finding $\angle BAD$:

   Since $ABCD$ is a cyclic quadrilateral, we know that opposite angles in a cyclic quadrilateral add up to $180^\circ$. Therefore: $\angle BAD + \angle BCD = 180^\circ$ Given $\angle BCD = 100^\circ$, we can solve for $\angle BAD$ as follows: $\angle BAD = 180^\circ - 100^\circ = 80^\circ$

2. Finding $\angle EAB$:

   Since $EF$ is a tangent to the circle at $A$, and $\angle DAF = 30^\circ$, we can apply the tangent-secant angle theorem. This theorem states that the angle between a tangent and a chord (in this case, $\angle EAB$) is equal to the angle in the alternate segment of the circle (which is $\angle DCA$).

   Here, $\angle DCA$ is equal to $\angle DAF = 30^\circ$, so: $\angle EAB = 30^\circ$

Final Answers:

• a) $\angle BAD = 80^\circ$
• b) $\angle EAB = 30^\circ$

Would you like more details on these solutions or have any questions?

Here are five related questions for further exploration:

1. What is the sum of opposite angles in any cyclic quadrilateral?
2. How does the tangent-secant angle theorem apply to other similar problems?
3. Can you find other angles in the quadrilateral $ABCD$?
4. What other properties hold true for cyclic quadrilaterals?
5. How would this problem change if $\angle BCD$ were different?

Tip: In circle geometry, understanding properties of cyclic quadrilaterals and tangent theorems is crucial for solving complex problems efficiently.