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 AB and 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 ⊙ P. All radii of a circle are congruent. Moreover, by the Congruent Central Angles Theorem, ∠ APB and ∠ CPD are congruent. AP ≅ BP ≅ CP ≅ DP [0.4em] ∠ APB ≅ ∠ CPD 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, AB and 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. AP ≅ BP ≅ CP ≅ DP Applying this information, and knowing that AB and 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 ∠ APB and ∠ CPD are congruent angles. △ APB ≅ △ CPD ⇓ ∠ APB ≅ ∠ CPD Notice that these angles are also central angles of the circle.

Therefore, by the Congruent Central Angles Theorem, it can be stated that AB and CD are congruent arcs. AB ≅ CD 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.