In the above circle, AB and CD are chords, and AB and CD are their corresponding arcs. Given that information and according to the theorem, the following relation holds true.
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 β³APB and β³CPD.
By the definition of a radius, the segments drawn to construct the triangles are radii of βP. All radii of a circle are congruent. Moreover, by the Congruent Central Angles Theorem, β APB and β CPD are congruent.
Since corresponding parts of congruent triangles are congruent, AB and CD are congruent segments. This proves the conditional statement.
ABβ CDβABβ CD
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 β³APB and β³CPD once again.
The segments drawn are all congruent because they are radii.
APβ BPβ CPβ DPβ
Applying this information, and knowing that AB and CD, are congruent, it can be stated that β³APB and β³CPD have three pairs of congruent sides.
By the Side-Side-Side Congruence Theorem, β³APB and β³CPD are congruent triangles. Since corresponding parts of congruent triangles are congruent, it can be stated that β APB and β CPD are congruent angles.
β³APBβ β³CPDββ APBβ β CPDβ
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.
ABβ CDβ
It has been proven that if two chords are congruent, then their corresponding arcs are also congruent.
ABβ CDβABβ CD
Having proven both the conditional statement and its converse completes the proof of the biconditional statement.