Question

here is a figure where ray ( r ) meets line ( ell ). the dashed rays are angle bisectors.

1. diego made the conjecture: \the angle formed between the angle bisectors is always a right angle, no matter what the angle between ( r ) and ( ell ) is.\ it is difficult to tell specifically which angles diego is talking about in his conjecture. label the diagram and rephrase diegos conjecture more precisely using your labels.

Explanation:

• First, label the angles. Let the angle between ray \(r\) and line \(\ell\) be \(\angle AOB\). Let the two angle - bisectors be \(OC\) (bisecting the angle above \(r\)) and \(OD\) (bisecting the angle below \(r\)).
• The sum of the two adjacent angles formed by ray \(r\) and line \(\ell\) is \(180^{\circ}\) (since they are supplementary). If \(OC\) bisects one of the angles (say \(\angle AOC=\frac{1}{2}\angle AOE\)) and \(OD\) bisects the other (\(\angle BOD = \frac{1}{2}\angle BOE\)), and \(\angle AOE+\angle BOE=180^{\circ}\).
• Then, using the angle - bisector property, the angle between the bisectors \(\angle COD=\frac{1}{2}(\angle AOE+\angle BOE)\)

Answer:

Let the angle between ray \(r\) and line \(\ell\) be composed of two adjacent angles \(\angle x\) and \(\angle y\) such that \(\angle x+\angle y = 180^{\circ}\). Let the bisectors of \(\angle x\) and \(\angle y\) be \(m\) and \(n\) respectively. Then the angle between \(m\) and \(n\) is a right angle. In other words, if \(m\) bisects \(\angle x\) (\(\angle1=\frac{1}{2}\angle x\)) and \(n\) bisects \(\angle y\) (\(\angle2=\frac{1}{2}\angle y\)), then \(\angle1+\angle2=\frac{1}{2}(\angle x + \angle y)=\frac{1}{2}\times180^{\circ}=90^{\circ}\)