Problem
GEO-B1-M01-P020 The Converse Problem with Parallels
#20
★★★☆☆ Level 3 of 5
Under the conditions of the previous problem, prove the converse: if \(AD\) bisects \(\angle EDF\), then \(AD\) bisects angle \(A\).
Again transfer the two angles at point \(D\) back to vertex \(A\).
Since \(DE\parallel AB\), the angle between \(DE\) and \(AD\) equals \(\angle BAD\). Since \(DF\parallel AC\), the angle between \(AD\) and \(DF\) equals \(\angle DAC\). By the condition, \(AD\) bisects \(\angle EDF\), so these two angles at \(D\) are equal. Therefore \(\angle BAD=\angle DAC\), meaning that \(AD\) bisects angle \(A\).
This problem reinforces the idea of equivalence after drawing parallel lines.