When dealing with circles and their angles, there are specific properties and theorems that help determine various measurements. One such property is related to the tangent-chord angle.
### Tangent-Chord Angle
A tangent-chord angle is formed when a tangent and a chord intersect at a point on a circle. This angle is known as a tangent-chord angle.
### Intercepted Arc
The intercepted arc is the portion of the circle that lies 'inside' the angle formed by a chord and a tangent. Essentially, it is the arc that is 'cut off' by the endpoints of the chord from the circle.
### Important Theorem
The key theorem to remember here is:
The measure of a tangent-chord angle is half the measure of the intercepted arc.
This theorem can be stated mathematically as:
[tex]\[ \text{Tangent-Chord Angle} = \frac{1}{2} \times \text{Intercepted Arc} \][/tex]
### Explanation
1. Suppose the tangent touches the circle at point [tex]\( A \)[/tex] and the chord intersects the tangent at point [tex]\( A \)[/tex] and extends to point [tex]\( B \)[/tex] on the circle.
2. Let [tex]\( C \)[/tex] be another point on the circle such that the arc [tex]\( BC \)[/tex] is intercepted by the chord [tex]\( AB \)[/tex].
3. The angle formed between the tangent line at [tex]\( A \)[/tex] and the chord [tex]\( AB \)[/tex] is our tangent-chord angle.
4. According to the theorem, this tangent-chord angle is equal to half of the measure of arc [tex]\( BC \)[/tex].
### Conclusion
Given this, if a statement asserts that the measure of a tangent-chord angle is half the measure of the intercepted arc, the statement is consistent with the established theorem.
Therefore, the answer to the question is:
A. True
