Based on the above diagram, the theorem can be written as follows.
BC≅DE⇔∠BAC≅∠DAE
Proof
The biconditional statement will be proved in two parts.
If Two Minor Arcs Are Congruent, Then the Corresponding Central Angles Are Congruent
Consider a circle and two congruent minor arcs.
By definition, the measure of a minor arc is equal to the measure of its related central angle. Since these two arcs are congruent, they have the same measure. Then, by the Transitive Property of Equality, the two central angles have the same measure.
⎩⎪⎪⎨⎪⎪⎧mBC=m∠BACmDE=m∠DAEmBC=mDE⇓m∠BAC=m∠DAE
Therefore, ∠BAC and ∠DAE are congruent angles.
∠BAC≅∠DAE✓
It has been proved that if two minor arcs are congruent, then their corresponding central angles are congruent.
If Two Central Angles Are Congruent, Then the Minor Arcs Are Congruent
Consider now a circle centered at point A. Let ∠BAC and ∠DAE be two congruent central angles.
By definition, the measure of a minor arc is equal to the measure of its related central angle. Since the central angles are congruent, they have the same measure. Again, by the Transitive Property of Equality, the two minor arcs have the same measure.
⎩⎪⎪⎨⎪⎪⎧mBC=m∠BACmDE=m∠DAEm∠BAC=m∠DAE⇓mBC=mDE
Consequently, since the arcs have the same measure, they are congruent.
BC≅DE✓
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