Congruent Central Angles Theorem

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.

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ m B C = m ∠ B A C m D E = m ∠ D A E m B C = m D E ⇓ m ∠ B A C = m ∠ D A E

Therefore,

∠ B A C

and

∠ D A E

are congruent angles.

$∠ B A C ≅ ∠ D A E ✓$

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 $∠ B A C$ and $∠ D A E$ 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.

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ m B C = m ∠ B A C m D E = m ∠ D A E m ∠ B A C = m ∠ D A E ⇓ m B C = m D E

Consequently, since the arcs have the same measure, they are congruent.

$B C ≅ D E ✓$

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Congruent Central Angles Theorem

Proof

If Two Minor Arcs Are Congruent, Then the Corresponding Central Angles Are Congruent

If Two Central Angles Are Congruent, Then the Minor Arcs Are Congruent

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