To find the measure of the indicated angle in the circle, let's analyze the given diagram and use the properties of circles and inscribed angles.

Diagram Analysis

1. Points and Angles:

   • The points $L, M, K,$ and $C$ lie on the circumference of the circle.
   • $\angle LKM$ is marked as $60^\circ$.
   • Arc $MC$ is $168^\circ$.
   • We need to find the measure of $\angle LMC$.

2. Properties of Circles:

   • An inscribed angle in a circle is half the measure of the arc it intercepts.
   • $\angle LKM$ intercepts arc $MC$.
   • $\angle LMC$ intercepts the arc opposite $MC$, which is arc $LK$.

Step 1: Calculate the Measure of Arc $LK$

Since the entire circle sums to $360^\circ$:

$\text{Arc } LK = 360^\circ - \text{Arc } MC$ $\text{Arc } LK = 360^\circ - 168^\circ$ $\text{Arc } LK = 192^\circ$

Step 2: Using the Inscribed Angle Theorem

The inscribed angle $\angle LMC$ intercepts arc $LK$. According to the inscribed angle theorem, the measure of an inscribed angle is half the measure of its intercepted arc:

$\angle LMC = \frac{1}{2} \times \text{Arc } LK$ $\angle LMC = \frac{1}{2} \times 192^\circ$ $\angle LMC = 96^\circ$

Conclusion

The measure of the indicated angle $\angle LMC$ is $96^\circ$.

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Would you like more details on this solution or have any further questions?

Here are some related questions:

1. How do you find the measure of an inscribed angle when given the arc it intercepts?
2. What is the relationship between a central angle and its intercepted arc in a circle?
3. How can you determine the measure of an arc given the inscribed angle?
4. What other properties of inscribed quadrilaterals are useful for solving circle problems?
5. How do properties of cyclic quadrilaterals help solve angle problems in circles?

Tip: Always remember that an inscribed angle is half the measure of its intercepted arc. This is a key property when working with circle geometry problems.