6. Secants, Tangents, and Angle Measures
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Let's start writing what Theorem 10.10 states.
Theorem 10.10 |
In a plane, a line is tangent to a circle if and only if it is perpendicular to a radius drawn to the point of tangency. |
The Inscribed Angle Theorem gives mFC=2m∠FAC. Then, by substitution we get m∠FAC+m∠CAE=90∘ and 2m∠FAC+mCA=180∘. Dividing the second equation by 2 and subtracting it from the first one gives m∠CAE−21mCA=0, which means that m∠CAE=21mCA.
The Inscribed Angle Theorem gives mCF=2m∠CAF. Then, by substitution we get m∠CAB=m∠CAF+90∘ and mCDA=2m∠CAF+180∘. Dividing the second equation by 2 and subtracting it from the first one gives m∠CAB−21mCDA=0, which means that m∠CAB=21mCDA.