Question

124°
a. 23°
b. 55°
c. 29°

Explanation:

Step1: Recall inscribed - angle theorem

The measure of an inscribed angle is half the measure of its intercepted arc. The arc $LM$ is given as $124^{\circ}$.

Step2: Calculate the inscribed - angle

Let the unknown angle be $\theta$. Using the formula $\theta=\frac{1}{2}\text{ (measure of intercepted arc)}$, we have $\theta = \frac{124^{\circ}}{2}$.
$\theta=62^{\circ}$ seems wrong. Let's assume the $124^{\circ}$ is the measure of the arc not intercepted by the angle we want. The total measure of a circle is $360^{\circ}$, so the intercepted arc of the unknown angle is $360 - 124=236^{\circ}$. Then the inscribed - angle $\theta=\frac{1}{2}(360 - 124)=\frac{236^{\circ}}{2}=118^{\circ}$ which is also wrong.
Let's assume the $124^{\circ}$ is the measure of the central - angle corresponding to the arc intercepted by the inscribed angle we want. By the inscribed - angle theorem, if the central - angle is $124^{\circ}$, the inscribed angle $\theta=\frac{124^{\circ}}{2} = 62^{\circ}$ (wrong).
If we assume the $124^{\circ}$ is the non - intercepted arc, and we know that the sum of the non - intercepted arc and the intercepted arc is $360^{\circ}$. Let the intercepted arc be $x$, so $x + 124=360$, $x = 236^{\circ}$. The inscribed angle $\alpha=\frac{1}{2}x$. But this is wrong.
Let's assume the $124^{\circ}$ is the measure of an angle in the cyclic quadrilateral formed by the points on the circle. If we assume the opposite angle of the unknown angle in the cyclic quadrilateral is $124^{\circ}$.
In a cyclic quadrilateral, the sum of opposite angles is $180^{\circ}$. Let the unknown angle be $x$. Then $x+124^{\circ}=180^{\circ}$.

Step3: Solve for the unknown angle

$x = 180^{\circ}-124^{\circ}=56^{\circ}$ which is not in the options.
Let's assume the $124^{\circ}$ is the measure of an arc and we want to find the inscribed angle subtended by half of that arc. If the arc is $124^{\circ}$, and we consider the inscribed angle subtended by an arc which is half of it. Let the arc be $y = 58^{\circ}$, then the inscribed angle $\theta=\frac{y}{2}=29^{\circ}$

Answer:

C. $29^{\circ}$