Problem
(8) Rachel says that the measure of \( x^{\circ} \) in the figure below is \( 30^{\circ} \). For Rachel to be correct, what condition must be true about the figure? (A) For \( x^{\circ}=30^{\circ} \), line segments Im and \( \overline{n o} \) must be parallel. (B) For \( x^{\circ}=30^{\circ} \), line segments \( \overline{l m} \) and \( \overline{n o} \) must be the same length. (C) For \( x^{\circ}=30^{\circ} \), line segment \( \overline{m n} \) must be the same length as \( \overline{l m} \) (D) There is no condition under which \( x \) could equal \( 30^{\circ} \).
Solution
Step 1 ：\(\angle AOB = 180^{\circ} - x^{\circ} \)
Step 2 ：\(\angle BON = 180^{\circ} - 150^{\circ} \)
Step 3 ：For \(x^{\circ}=30^{\circ}\), \(\angle AOB = 180^{\circ} - 30^{\circ} \) and \(\angle BON = 180^{\circ} - 150^{\circ}\), so \(\angle AOB = \angle BON\)
Step 4 ：Corresponding angles are equal, therefore line segments \(\overline{lm}\) and \(\overline{no}\) must be parallel
