Problem
GEO-B1-M08-P021 One Pair Equal and Parallel
#21
★★★★☆ Level 4 of 5
In quadrilateral \(ABCD\), it is known that \(AB\parallel CD\) and \(AB=CD\). Prove that \(ABCD\) is a parallelogram.
Draw diagonal \(AC\) and look for congruent triangles.
Draw \(AC\). Since \(AB\parallel CD\), \(\angle BAC=\angle ACD\). Also \(AB=CD\), and \(AC\) is common. Therefore \(\triangle BAC\cong\triangle DCA\). Hence \(\angle BCA=\angle CAD\), so \(BC\parallel AD\). Together with \(AB\parallel CD\), this gives a parallelogram.
Checks the choice of a diagonal as an auxiliary line.