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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.

$AB≅CD⇔AB≅CD$

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

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∠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.

$AB≅CD⇒AB≅CD$

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. $AB≅CD⇒AB≅CD$

Having proven both the conditional statement and its converse completes the proof of the biconditional statement.

$AB≅CD⇔AB≅CD$