To solve the problem, we need to determine the size of angle $\angle AED$. Here’s the step-by-step solution:

Step 1: Analyze the geometry

• The tangents $AB$ and $BD$ meet the circle at points $A$ and $D$, respectively.
• $C$ is the center of the circle.
• $\angle ABD = 42^\circ$ is given.
• $AE$ and $DE$ are chords of the circle.

Step 2: Use the tangent-segment angle rule

The angle between a tangent and a chord drawn at the point of contact is equal to the angle subtended by the chord on the opposite side of the circle.

Thus: $\angle BAE = \angle ADE$

This means: $\angle ADE = 42^\circ$

Step 3: Find $\angle AED$

In the quadrilateral $AEDC$, the sum of opposite angles is $180^\circ$ (cyclic quadrilateral property). Therefore: $\angle AED + \angle ADE = 180^\circ$

Substitute $\angle ADE = 42^\circ$: $\angle AED + 42^\circ = 180^\circ$

Solve for $\angle AED$: $\angle AED = 180^\circ - 42^\circ = 138^\circ$

Final Answer:

$\boxed{\angle AED = 138^\circ}$

Let me know if you'd like more clarification or have additional questions! Here are 5 follow-up questions for practice:

1. What is the cyclic quadrilateral property, and how is it applied in this problem?
2. Can you explain why $\angle BAE = \angle ADE$ in detail?
3. How does the tangent-segment angle rule apply to circles?
4. If $\angle ABD$ were different, how would that change the calculation?
5. Can you identify other properties of cyclic quadrilaterals in this setup?

Tip: Always check for cyclic quadrilateral properties when working with inscribed angles and opposite angles.