Given: A circle with inscribed quadrilateral ABCD
Prove: ∠A and ∠C are supplementary.

✓ 1. Let m#& \overset{\frown}{\sf DCB}#& = a°
Then m#& \overset{\frown}{\sf DAB}#& = 90 - a 180 - a 360 - a .

✓ 2. By the inscribed angle theorem, m∠A = a/2 a 2a .

✓ 3. Also by the inscribed angle theorem, m∠C = 180 - a/2 360 - a/2 (360 - a)/2 .

✓ 4. Therefore, m∠A + m∠C = #&\frac{a}{\sf 2} + \frac{{\sf 360} - a}{\sf 2}#&,
which simplifies to 90 180 360 .

✓ 5. Therefore, m∠A and m∠C are complementary supplementary by the definition of supplementary angles.