Question

enter the given angle measure into the diagram.
given:

• (overline{iq}) bisects (angle xir).
• (mangle xir = 120^{circ}).

enter each measure required to calculate (mangle qij), starting with
the given information.
the measure of (angle qij) can be calculated.
(mangle qij = 120^{circ})
the measure of (angle qij) can not be calculated.
diagram notes:

Explanation:

Step1: Use the angle - bisector property

Since \( \overline{IQ} \) bisects \( \angle XIR \), then \( \angle XIQ=\angle QIR \). Given \( m\angle XIR = 120^{\circ} \), by the angle - bisector formula \( \angle XIQ=\angle QIR=\frac{1}{2}\angle XIR \).

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

Step2: Use the linear - pair property

\( \angle QIJ\) and \( \angle XIQ\) form a linear pair. The sum of angles in a linear pair is \( 180^{\circ}\). Let \( m\angle QIJ = x\). Then \( x+\angle XIQ = 180^{\circ}\).
Substitute \( \angle XIQ = 60^{\circ}\) into the equation:

$$ x+60^{\circ}=180^{\circ} $$

$$ x=180^{\circ}- 60^{\circ} $$

$$ x = 120^{\circ} $$

Answer:

\( m\angle QIJ=120^{\circ}\)