Congruent Corresponding Chords Theorem

Since the theorem is a biconditional statement, the proof will consist of two parts, the conditional statement and its converse.

Part I: If Two Minor Arcs Are Congruent, Then Their Corresponding Chords Are Also Congruent

To prove the conditional statement, assume that $\overgroup{AB}$ and $\overgroup{CD}$ are congruent arcs. Then, consider $△APB$ and $△CPD.$ By the definition of a radius, the segments drawn to construct the triangles are radii of $\odot P.$ All radii of a circle are congruent. Moreover, by the Congruent Central Angles Theorem, $\angle APB$ and $\angle CPD$ are congruent. Considering the above congruences, by the Side-Angle-Side Congruence Theorem, $△APB$ and $△CPD$ are congruent triangles.

Since corresponding parts of congruent triangles are congruent, $\overline{AB}$ and $\overline{CD}$ 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 $△APB$ and $△CPD$ once again. The segments drawn are all congruent because they are radii. Applying this information, and knowing that $\overline{AB}$ and $\overline{CD},$ are congruent, it can be stated that $△APB$ and $△CPD$ have three pairs of congruent sides. By the Side-Side-Side Congruence Theorem, $△APB$ and $△CPD$ are congruent triangles. Since corresponding parts of congruent triangles are congruent, it can be stated that $\angle APB$ and $\angle CPD$ are congruent angles. Notice that these angles are also central angles of the circle. Therefore, by the Congruent Central Angles Theorem, it can be stated that $\overgroup{AB}$ and $\overgroup{CD}$ 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.