Question

lines e and f are parallel. the m∠9 = 80° and m∠5 = 55°. which angle measures are correct? choose three correct answers. m∠8 = 55° m∠3 = 55° m∠2 = 125° m∠14 = 100° m∠12 = 100°

Explanation:

Step1: Vertical Angles and Linear Pairs

Vertical angles are equal. Linear pairs sum to \(180^{\circ}\). Given \(m\angle5 = 55^{\circ}\), \(\angle5\) and \(\angle8\) are vertical angles. So \(m\angle8=m\angle5 = 55^{\circ}\). \(\angle3\) and \(\angle5\) are alternate - interior angles (since lines \(e\) and \(f\) are parallel). So \(m\angle3=m\angle5 = 55^{\circ}\). \(\angle2\) and \(\angle4\) form a linear pair. \(\angle4\) and \(\angle5\) are same - side interior angles. \(m\angle4=180 - 55=125^{\circ}\), then \(m\angle2=m\angle4 = 125^{\circ}\) (vertical angles).

Step2: Angles related to \(\angle9\)

\(\angle9\) and \(\angle11\) are vertical angles (\(m\angle11 = 80^{\circ}\)). \(\angle11\) and \(\angle12\) form a linear pair (\(m\angle12=180 - 80=100^{\circ}\)). \(\angle12\) and \(\angle14\) are alternate - interior angles (since lines \(e\) and \(f\) are parallel). So \(m\angle14=m\angle12 = 100^{\circ}\).

Answer:

\(m\angle8 = 55^{\circ}\), \(m\angle3 = 55^{\circ}\), \(m\angle2 = 125^{\circ}\), \(m\angle14 = 100^{\circ}\), \(m\angle12 = 100^{\circ}\)