Construct inscribed/circumscribed circles of triangles and prove angle properties of cyclic quadrilaterals.
Determine whether a quadrilateral can be cyclic from its opposite angle pairs
Problem
Determine whether a quadrilateral can be cyclic if \(\text{one}~\text{pair}~\text{of}~\text{opposite}~\text{angles}\) is \(100^{\circ}\) and \(80^{\circ}\), and the other pair is \(70^{\circ}\) and \(110^{\circ}\).
Testing whether a quadrilateral can be cyclic uses the converse of the opposite-angle theorem. We’ll preserve the vertex order, form the two opposite pairs, and check each sum against a straight angle rather than pairing adjacent angles.
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