Question

look at this diagram: if \\( \overleftrightarrow { p r } \\) and \\( \overleftrightarrow { s u } \\) are parallel lines and \\( m \angle u t v = 54 ^ { \circ } \\), what is \\( m \angle r q o \\)?

Explanation:

Step1: Identify the relationship between angles

Since \( \overleftrightarrow{PR}\) and \( \overleftrightarrow{SU}\) are parallel lines and \(VO\) is a transversal. \( \angle UTV\) and \( \angle RQT\) are vertical - angle. So \(m\angle RQT=m\angle UTV = 54^{\circ}\).

Step2: Use the linear - pair relationship

\( \angle RQT\) and \( \angle RQO\) form a linear pair. The sum of angles in a linear pair is \(180^{\circ}\). So \(m\angle RQT + m\angle RQO=180^{\circ}\).

Step3: Solve for \(m\angle RQO\)

Substitute \(m\angle RQT = 54^{\circ}\) into the equation \(m\angle RQT + m\angle RQO=180^{\circ}\). Then \(m\angle RQO=180^{\circ}-m\angle RQT\).

$$m\angle RQO = 180 - 54$$

$$m\angle RQO=126^{\circ}$$

Answer:

\(126\)