Problem
GEO-B1-M04-P006 An Equal and Parallel Pair of Sides
#6
★★☆☆☆ Level 2 of 5
In a convex quadrilateral \(ABCD\), suppose \(AB\parallel CD\) and \(AB=CD\). Prove that \(ABCD\) is a parallelogram.
Draw diagonal \(AC\) and prove that \(BC\parallel AD\).
Draw \(AC\). Since \(AB\parallel CD\), \(\angle BAC=\angle ACD\). Also \(AB=CD\), and \(AC\) is common. Thus \(\triangle BAC\cong\triangle DCA\) by SAS. Therefore \(\angle BCA=\angle CAD\), so \(BC\parallel AD\). Together with \(AB\parallel CD\), this gives a parallelogram.
One of the main parallelogram criteria; encourage a careful proof through triangles.