Question

the measure of $overparen{axb}$ is $234^{circ}$. what is the measure of $angle abc$, the tangent - chord angle?

Explanation:

Step1: Recall tangent - chord angle formula

The measure of a tangent - chord angle is half the measure of its intercepted arc.

Step2: Find the measure of the intercepted arc

The intercepted arc of $\angle ABC$ is the minor arc $\overset{\frown}{AB}$. The major arc $\overset{\frown}{AXB}=234^{\circ}$, and the sum of the measures of a major and minor arc of a circle is $360^{\circ}$. So the measure of the minor arc $\overset{\frown}{AB}=360^{\circ}- 234^{\circ}=126^{\circ}$.

Step3: Calculate the measure of $\angle ABC$

Since the measure of $\angle ABC$ (tangent - chord angle) is half the measure of its intercepted arc, and the intercepted arc $\overset{\frown}{AB}$ has a measure of $126^{\circ}$, then $m\angle ABC = 126^{\circ}$.

Answer:

A. $126^{\circ}$