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{{ printedBook.courseTrack.name }} {{ printedBook.name }} In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding central angles are congruent.

Based on the above diagram, the theorem can be written as follows.

$BC≅DE⇔∠BAC≅∠DAE$

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. 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. Then, by the Transitive Property of Equality, the two minor arcs have the same measure. $⎩⎪⎪⎨⎪⎪⎧ mBC=m∠BACmDE=m∠DAEm∠BAC=m∠DAE ⇓mBC=mDE $ Therefore, by definition of congruent arcs, $BC$ and $DE$ are congruent.

$BC≅DE$