Problem
GEO-B1-M02-P018 Two Equal Pairs of Sides
#18
★★★☆☆ Level 3 of 5
In quadrilateral \(ABCD\), \(AB=AD\) and \(CB=CD\). Prove that diagonal \(AC\) bisects angles \(A\) and \(C\).
Compare triangles \(ABC\) and \(ADC\).
In triangles \(ABC\) and \(ADC\), we have \(AB=AD\), \(CB=CD\), and common side \(AC\). By SSS, the triangles are congruent. Therefore \(\angle BAC=\angle CAD\) and \(\angle BCA=\angle ACD\). Hence \(AC\) bisects angles \(A\) and \(C\).
An entry point to the idea of an axis of symmetry without formally introducing it.