a. 26°

Free Solution from UpStudy:

To find the measure of angle \(\angle LMK\) in the circle, we need to use the properties of inscribed angles. In a circle, an inscribed angle is half the measure of the intercepted arc.

Given:

- \(\angle LKM = 52^ \circ \)

\(\angle LMK\) is an inscribed angle that intercepts the same arc as \(\angle LKM\). According to the inscribed angle theorem, the measure of an inscribed angle is half the measure of its intercepted arc.

Therefore, the measure of \(\angle LMK\) is:

\(\angle LMK = \frac { 1} { 2} \times 52^ \circ = 26^ \circ \)

Supplemental Knowledge:

The Inscribed Angle Theorem is a fundamental concept in circle geometry. It states that an inscribed angle is always half the measure of the intercepted arc. An inscribed angle is formed when two chords of a circle intersect on the circle's circumference. If an inscribed angle intercepts an arc measuring \(m\) degrees, then the inscribed angle measures \(\frac { m} { 2} \) degrees. This relationship allows us to determine the measure of inscribed angles based on the arcs they intercept.

In this problem, given \(\angle LKM = 52^ \circ \), and since \(\angle LMK\) intercepts the same arc, the measure of \(\angle LMK\) is:

\(\angle LMK = \frac { 1} { 2} \times 52^ \circ = 26^ \circ \)

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