Two chords in the same circle are congruent if and only if the associated central angles are congruent.A.TrueB.FalseSUBMITarrow_backPREVIOUS

Break Down the Problem

1. Identify the properties of chords and central angles in a circle.
2. Determine the relationship between congruent chords and their associated central angles.

Relevant Concepts

• In a circle, if two chords are congruent, their corresponding central angles are also congruent.
• Conversely, if two central angles are congruent, the chords they subtend are also congruent.

Analysis and Detail

1. Chords and Central Angles: For any given circle, the length of a chord is directly related to the measure of the central angle subtended by the chord at the center of the circle. Consequently, if the lengths of the chords are equal (congruent), then the angles are equal (congruent).
2. Logical Argument: This relationship can be stated as:
   • If $AB \cong CD$ (chords), then $m\angle AOB \cong m\angle COD$ (central angles).
   • If $m\angle AOB \cong m\angle COD$ (central angles), then $AB \cong CD$ (chords).

Verify and Summarize

Both statements hold true; therefore, the assertion that two chords are congruent if and only if their corresponding central angles are congruent is accurate.

Final Answer

A. True