To determine the truth of the statement "Two arcs of a circle are congruent if and only if their associated chords are congruent," let's analyze the concepts involved.
1. Arcs of a Circle: An arc is a portion of the circumference of a circle. Two arcs are said to be congruent if they have the same length.
2. Chords of a Circle: A chord is a straight line segment whose endpoints both lie on the circle. The length of the chord is determined by the distance between these endpoints.
3. Relationship between Arcs and Chords:
- If two arcs are congruent, they subtend angles of the same measure at the center of the circle.
- The length of an arc is directly related to the angle it subtends at the center of the circle.
- For congruent arcs, these subtended angles are equal, hence their corresponding chords, which span these angles, will also be of equal length.
Therefore, if two arcs of a circle are congruent, their corresponding chords must also be equal in length. Conversely, if two chords in a circle are of equal length, the arcs that subtend these chords will also be congruent because they subtend equal angles at the center of the circle.
Thus, the statement "Two arcs of a circle are congruent if and only if their associated chords are congruent" is indeed true.
So, the correct answer is:
- A. True
