Question - Circular Geometry Problem Involving Inscribed Angles

We are given a circle with inscribed angles and we need to determine the value of angle \( x \).

Firstly, we notice that angle \( KLN \) is an inscribed angle that subtends arc \( KM \). According to the inscribed angle theorem, an inscribed angle is half the measure of its subtended arc. Therefore, the measure of arc \( KM \) is \( 120^\circ \times 2 = 240^\circ \).

Since angle \( LNM \) is also an inscribed angle subtending the arc \( LM \), and it measures \( 110^\circ \), the measure of arc \( LM \), by the inscribed angle theorem, is \( 110^\circ \times 2 = 220^\circ \).

Now arc \( LM \) and arc \( KM \) overlap to form the entire circumference of the circle, which measures \( 360^\circ \). Therefore, the remaining arc, \( KL \), must have a measure of \( 360^\circ - 220^\circ = 140^\circ \).

Since angle \( x \), which is \( \angle KMN \), is an inscribed angle that subtends arc \( KL \), by the inscribed angle theorem, \( x \) is half the measure of arc \( KL \), hence:

\[ x = \frac{140^\circ}{2} = 70^\circ \]

The answer is \( A: 70^\circ \).