Question

the measure of \\(\widehat{axb}\\) is \\(196^\circ\\). what is the measure of \\(\angle abc\\), the tangent-chord angle?

a. \\(196^\circ\\)
b. \\(96^\circ\\)
c. \\(106^\circ\\)
d. \\(98^\circ\\)

Explanation:

Identify the given geometric elements

We are given a circle with a chord \(AB\) and a tangent line passing through point \(B\). Point \(C\) lies on this tangent line. The angle \(\angle ABC\) is formed by the tangent line segment \(BC\) and the chord \(AB\). We are also given that the measure of the intercepted arc \(\widehat{AXB}\) is \(196^\circ\).

Apply the Tangent-Chord Theorem

The Tangent-Chord Theorem states that the measure of an angle formed by a tangent and a chord that intersect at a point on a circle is exactly half the measure of its intercepted arc.

Calculate the angle measure

Using the theorem, we set up the relationship:

$$ m\angle ABC = \frac{1}{2} \cdot m\widehat{AXB} $$

Substitute the given arc measure:

$$ m\angle ABC = \frac{1}{2} \cdot 196^\circ = 98^\circ $$

This matches option D.

Answer:

• (A) \(196^\circ\)
• (B) \(96^\circ\)
• (C) \(106^\circ\)
• (D) \(98^\circ\) (Correct answer)