Problem
GEO-B1-M01-P018 Two Pairs of Parallel Lines
#18
★★★☆☆ Level 3 of 5
The diagonals of quadrilateral \(ABCD\) intersect at \(O\). It is known that \(\angle BAC=\angle DCA\) and \(\angle BCA=\angle DAC\). Prove that \(AB\parallel CD\) and \(BC\parallel AD\).
Consider line \(AC\) as a transversal for each pair of lines.
Angles \(\angle BAC\) and \(\angle DCA\) are alternate interior angles for lines \(AB\) and \(CD\) with transversal \(AC\). They are equal, so \(AB\parallel CD\). Similarly, angles \(\angle BCA\) and \(\angle DAC\) are alternate interior angles for lines \(BC\) and \(AD\) with transversal \(AC\). They are equal, hence \(BC\parallel AD\).
Although diagonals are mentioned, the solution is entirely angle-based.