To solve the problem in the diagram, we can use the fact that the quadrilateral $IJKL$ is inscribed in the circle, making it a cyclic quadrilateral. One of the key properties of a cyclic quadrilateral is that the opposite angles are supplementary, i.e., their sum equals $180^\circ$.

Step 1: Identify the given angles

• $\angle I = 66^\circ$
• $\angle J = 99^\circ$

Step 2: Apply the cyclic quadrilateral property

The opposite angles in a cyclic quadrilateral add up to $180^\circ$:

1. $\angle I + \angle K = 180^\circ$ Substituting $\angle I = 66^\circ$: $66^\circ + \angle K = 180^\circ$ $\angle K = 180^\circ - 66^\circ = 114^\circ$

2. $\angle J + \angle L = 180^\circ$ Substituting $\angle J = 99^\circ$: $99^\circ + \angle L = 180^\circ$ $\angle L = 180^\circ - 99^\circ = 81^\circ$

Final Answer:

The measure of $\angle K$ is: $\boxed{114^\circ}$

Let me know if you'd like further explanation or assistance with related problems. Here are five related questions to explore this concept further:

1. What is the measure of $\angle L$ in this diagram?
2. Why do opposite angles in a cyclic quadrilateral sum to $180^\circ$?
3. How would the problem change if quadrilateral $IJKL$ were not cyclic?
4. Can you derive the relationship between the arcs subtended by the angles $\angle I$ and $\angle K$?
5. What is the relationship between the perpendicular bisectors of $IJKL$ and the center of circle $M$?

Tip:

Always remember the properties of cyclic quadrilaterals when working with problems involving inscribed figures. This can simplify angle calculations significantly!