Problem
GEO-B1-M01-P007 Bisectors of Same-Side Interior Angles
#7
★★☆☆☆ Level 2 of 5
Two parallel lines are cut by a third line. Prove that the bisectors of two same-side interior angles are perpendicular.
The same-side interior angles sum to \(180^\circ\). What happens after each angle is halved?
Let the same-side interior angles be \(\alpha\) and \(\beta\). Then \(\alpha+\beta=180^\circ\). Their bisectors make angles \(\frac{\alpha}{2}\) and \(\frac{\beta}{2}\) with the transversal. The sum of these angles is \(\frac{\alpha+\beta}{2}=90^\circ\). Therefore the angle between the bisectors is \(90^\circ\), so they are perpendicular.
A good problem for moving from a fact to a general symbolic argument.