We are given a circle with an arc $KL$ measuring $111^\circ$. We need to find the measure of angle $KLM$, where lines $KL$ and $LM$ appear to be tangent to the circle. The lines that appear to be tangent are indeed tangent.

1. Problem Description

We are given a circle with an arc $KL$ measuring $111^\circ$. We need to find the measure of angle $KLM$, where lines $KL$ and $LM$ appear to be tangent to the circle. The lines that appear to be tangent are indeed tangent.

2. Solution Steps

The angle $KLM$ is formed by two tangents to the circle. Let the measure of arc $KL$ be $x$. The measure of the major arc $KL$ is $360^\circ - x$.

The angle formed by two tangents is half the difference of the intercepted arcs.

In this case, $x = 111^\circ$.

Then the measure of the major arc $KL$ is $360^\circ - 111^\circ = 249^\circ$.

The measure of $\angle KLM$ is half the difference between the measures of the intercepted arcs:

$m\angle KLM = \frac{1}{2} |(360^\circ - 111^\circ) - 111^\circ| = \frac{1}{2} |249^\circ - 111^\circ| = \frac{1}{2} |138^\circ| = \frac{1}{2} (138^\circ) = 69^\circ$.

3. Final Answer

69.0