Problem
GEO-B2-M01-P010 An Exterior Angle of a Cyclic Quadrilateral
#10
★★★☆☆ Level 3 of 5
In cyclic quadrilateral \(ABCD\), lines \(AD\) and \(BC\) meet at \(P\). Prove that \(\angle APB\equiv\angle DAB+\angle ABC\pmod{180^\circ}\).
Write the angle between lines \(AD\) and \(BC\) as the difference of angles measured from line \(AB\).
Angle \(\angle APB\) is the oriented angle between lines \(AD\) and \(BC\). Rotate from \(AD\) to \(AB\), and then from \(AB\) to \(BC\). We get \(\angle APB\equiv\angle(AD,AB)+\angle(AB,BC)\). But \(\angle(AD,AB)=\angle DAB\), and \(\angle(AB,BC)\equiv\angle ABC\pmod{180^\circ}\). Therefore \(\angle APB\equiv\angle DAB+\angle ABC\pmod{180^\circ}\).
This problem trains the algebra of oriented angles rather than a new circle fact.