Question

19.2: that cant be right, can it?
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.
2. is the conjecture true? explain your reasoning.

Explanation:

1. Let's label the two adjacent angles formed by ray \(r\) and line \(\ell\) as \(\angle 1\) and \(\angle 2\). Let the angle bisectors of \(\angle 1\) and \(\angle 2\) be \(b_1\) and \(b_2\) respectively.

Diego's conjecture can be rephrased as: "The angle formed between the bisectors of two adjacent angles that form a linear - pair (the angles between ray \(r\) and line \(\ell\)) is always a right angle."

1. # Explanation:

Step1: Use the property of a linear - pair and angle bisectors

We know that if two angles \(\angle 1\) and \(\angle 2\) form a linear - pair, then \(\angle 1+\angle 2 = 180^{\circ}\).
Let the measure of \(\angle 1=x\) and the measure of \(\angle 2 = y\), so \(x + y=180^{\circ}\).
If \(b_1\) is the bisector of \(\angle 1\), then the measure of the angle formed by \(b_1\) and the common side of \(\angle 1\) and \(\angle 2\) is \(\frac{x}{2}\).
If \(b_2\) is the bisector of \(\angle 2\), then the measure of the angle formed by \(b_2\) and the common side of \(\angle 1\) and \(\angle 2\) is \(\frac{y}{2}\).

Step2: Calculate the measure of the angle between the bisectors

The measure of the angle between the two bisectors \(b_1\) and \(b_2\) is \(\frac{x + y}{2}\).
Since \(x + y = 180^{\circ}\), then \(\frac{x + y}{2}=\frac{180^{\circ}}{2}=90^{\circ}\)

Answer:

1. Label the two adjacent angles formed by \(r\) and \(\ell\) as \(\angle 1\) and \(\angle 2\). Diego's conjecture: "The angle between the bisectors of two adjacent angles (that form a linear - pair) is always a right angle."
2. The conjecture is true. Because if two adjacent angles \(\angle 1\) and \(\angle 2\) form a linear - pair (\(\angle 1+\angle 2 = 180^{\circ}\)), and their bisectors divide them into \(\frac{\angle 1}{2}\) and \(\frac{\angle 2}{2}\), the angle between the bisectors is \(\frac{\angle 1+\angle 2}{2}=\frac{180^{\circ}}{2} = 90^{\circ}\)