Problem
GEO-B1-M02-P015 Diagonals Bisect Each Other
#15
★★★☆☆ Level 3 of 5
Segments \(AC\) and \(BD\) intersect at \(O\) and are bisected by this point: \(AO=OC\), \(BO=OD\). Prove that \(AB\parallel CD\) and \(AD\parallel BC\).
Prove that triangles \(AOB\) and \(COD\) are congruent, then use angles.
Angles \(AOB\) and \(COD\) are vertical, so they are equal. Together with \(AO=OC\) and \(BO=OD\), this gives \(\triangle AOB=\triangle COD\) by SAS. Hence \(\angle ABO=\angle CDO\), and equal alternate interior angles imply \(AB\parallel CD\). Similarly, from congruent triangles \(AOD\) and \(COB\), we get \(AD\parallel BC\).
Connects triangle congruence with the parallelism criterion from Module 1.