Problem
GEO-B1-M07-P025 Hidden Parallelogram From One Pair of Sides
#25
★★★★☆ Level 4 of 5
In quadrilateral \(ABCD\), it is known that \(AB=CD\) and \(AB\parallel CD\). Draw diagonal \(AC\) and prove that \(ABCD\) is a parallelogram.
After drawing the diagonal, look for congruent triangles.
Draw \(AC\). Since \(AB\parallel CD\), we have \(\angle BAC=\angle ACD\). Also \(AB=CD\), and \(AC\) is common. Therefore \(\triangle BAC\cong\triangle DCA\) by two sides and the included angle. Hence \(\angle BCA=\angle CAD\), so \(BC\parallel AD\). Thus both pairs of opposite sides are parallel, and \(ABCD\) is a parallelogram.
A diagonal turns a quadrilateral problem into a congruent-triangles problem.