Question

the measure of (overarc{ab}) is (126^circ). what is the measure of (angle abc)?
options: (152^circ), (126^circ), (68^circ), (63^circ)

Explanation:

Step1: Recall the tangent - chord angle theorem

The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. Here, \(BC\) is a tangent to the circle at point \(B\), and \(AB\) is a chord. The intercepted arc is \(\overset{\frown}{AB}\) with measure \(126^{\circ}\).

Step2: Apply the theorem to find \(\angle ABC\)

Let \(m\angle ABC\) be the measure of the angle formed by the tangent \(BC\) and the chord \(AB\). According to the tangent - chord angle theorem, \(m\angle ABC=\frac{1}{2}m\overset{\frown}{AB}\).
Substitute \(m\overset{\frown}{AB} = 126^{\circ}\) into the formula: \(m\angle ABC=\frac{1}{2}\times126^{\circ}=63^{\circ}\).

Answer:

\(63^{\circ}\) (corresponding to the option with \(63^{\circ}\))