Question

enter the given angle measure into the diagram.
given:

• iq bisects ∠xir.
• m∠xir = 120°.

enter each measure required to calculate m∠qij, starting with
the given information.
the measure of ∠quj can be calculated.
m∠quj = 120°
the measure of ∠quj can not be calculated.
diagram notes:

Explanation:

Step1: Use the angle - bisector property

Since \( \overrightarrow{IQ}\) bisects \( \angle XIR\), and \(m\angle XIR = 120^{\circ}\), then \(m\angle QIR=\frac{1}{2}m\angle XIR\).

$$m\angle QIR=\frac{1}{2}\times120^{\circ} = 60^{\circ}$$

Step2: Use the linear - pair or angle - sum property

We know that \(m\angle QIJ + m\angle QIR+m\angle RIJ = 180^{\circ}\) (a straight - line angle). Given \(m\angle QIJ = 120^{\circ}\) and \(m\angle QIR = 60^{\circ}\), we substitute into the equation:

$$120^{\circ}+60^{\circ}+m\angle RIJ=180^{\circ}$$

$$m\angle RIJ=180^{\circ}-(120^{\circ} + 60^{\circ})$$

$$m\angle RIJ = 0^{\circ}$$

This is wrong. Let's use another approach.

We know that \(m\angle QIJ\) and \(m\angle JIR\) are related. Since \(m\angle XIR = 120^{\circ}\) and \( \overrightarrow{IQ}\) bisects \( \angle XIR\), \(m\angle QIR = 60^{\circ}\). Also, \(m\angle QIJ=120^{\circ}\).

We use the formula \(m\angle RIJ=m\angle QIJ - m\angle QIR\) (assuming the correct angle - relationship from the diagram)

$$m\angle RIJ=120^{\circ}-60^{\circ}$$

Answer:

\(60\)