Circumscribed Angle Theorem - Rule, Proof and Examples

By definition, a circumscribed angle is an angle whose sides are tangents to a circle. Since ∠ ADB is a circumscribed angle, DA and DB are tangents to ⊙ C at points A and B, respectively. By the Tangent to Circle Theorem, CA is perpendicular to DA and CB is perpendicular to DB.

Notice that ADBC is a quadrilateral and two of its angles are right angles. Recall that the sum of all of the angles in a quadrilateral is 360^(∘). Substituting the known angle measures and solving for m∠ ADB will give the desired equation.

m∠ ADB + m∠ DAC + m∠ ACB + m∠ CBD = 360^(∘)

SubstituteII

m∠ DAC= 90^(∘), m∠ CBD= 90^(∘)

m∠ ADB + 90^(∘) + m∠ ACB + 90^(∘) = 360^(∘)

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Solve for m∠ ADB

AddTerms

Add terms

m∠ ADB + m∠ ACB + 180 ^(∘) = 360^(∘)

SubEqn

LHS-180^(∘)=RHS-180^(∘)

m∠ ADB+ m∠ ACB = 180 ^(∘)

SubEqn

LHS-m∠ ACB=RHS-m∠ ACB

m∠ ADB = 180 ^(∘) - m∠ ACB

This completes the proof.

Proof

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