Since the theorem is a , the proof will consist of two parts, the statement and its .
Part I: If Two Minor Arcs Are Congruent, Then Their Corresponding Chords Are Also Congruent
To prove the conditional statement, assume that
are congruent arcs. Then, consider
By the definition of a , the segments drawn to construct the triangles are radii of
All radii of a circle are congruent. Moreover, by the ,
Considering the above congruences, by the ,
are congruent triangles.
Since corresponding parts of congruent triangles are congruent, and are congruent segments. This proves the conditional statement.
Part II: If Two Chords Are Congruent, Then Their Corresponding Arcs Are Also Congruent
Next, the converse of the above statement will be proven. Consider
The segments drawn are all congruent because they are radii.
Applying this information, and knowing that
are congruent, it can be stated that
have three pairs of congruent sides.
By the ,
are congruent triangles. Since corresponding parts of congruent triangles are congruent, it can be stated that
are congruent .
Notice that these angles are also of the circle.
Therefore, by the Congruent Central Angles Theorem, it can be stated that
are congruent arcs.
It has been proven that if two chords are congruent, then their corresponding arcs are also congruent.
Having proven both the conditional statement and its converse completes the proof of the biconditional statement.