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Rule

Congruent Corresponding Chords Theorem

Two minor arcs are congruent if and only if their corresponding chords are congruent in the same circle congruent circles.
Congruent arcs and their corresponding chords

In the above circle, and are chords, and and are their corresponding arcs. Given that information and according to the theorem, the following relation holds true.

Proof

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 and are congruent arcs. Then, consider and
The triangles APB and CPD are shown on the circle
By the definition of a radius, the segments drawn to construct the triangles are radii of All radii of a circle are congruent. Moreover, by the Congruent Central Angles Theorem, and are congruent.
Considering the above congruences, by the Side-Angle-Side Congruence Theorem, and are congruent triangles.
The triangles APB and CPD are shown on the circle

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 and once again.
The triangles APB and CPD are shown on the circle
The segments drawn are all congruent because they are radii.
Applying this information, and knowing that and are congruent, it can be stated that and have three pairs of congruent sides.
By the Side-Side-Side Congruence Theorem, and are congruent triangles. Since corresponding parts of congruent triangles are congruent, it can be stated that and 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 and 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.

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