Question

lines e and f are parallel. the m∠9 = 80° and m∠5 = 95°. which angle measures are correct? select three options. □m∠2 = 125° □m∠3 = 55° □m∠8 = 55° □m∠12 = 100° □m∠14 = 100°

Explanation:

Step1: Use linear - pair and vertical - angle relationships

Since \(m\angle9 = 80^{\circ}\), and \(\angle9\) and \(\angle11\) are vertical angles, \(m\angle11=80^{\circ}\). Also, \(\angle9\) and \(\angle10\) form a linear pair (\(\angle9+\angle10 = 180^{\circ}\)), so \(m\angle10 = 100^{\circ}\). And \(\angle10\) and \(\angle12\) are vertical angles, so \(m\angle12=100^{\circ}\).
Since \(m\angle5 = 55^{\circ}\), and \(\angle5\) and \(\angle7\) are vertical angles, \(m\angle7 = 55^{\circ}\). Also, \(\angle5\) and \(\angle3\) are alternate - interior angles (because \(e\parallel f\)), so \(m\angle3 = 55^{\circ}\). And \(\angle3\) and \(\angle8\) are alternate - exterior angles (because \(e\parallel f\)), so \(m\angle8 = 55^{\circ}\).

Step2: Check each option

• For \(m\angle2\): \(\angle2\) and \(\angle3\) form a linear pair (\(\angle2+\angle3=180^{\circ}\)). If \(\angle3 = 55^{\circ}\), then \(m\angle2=180 - 55=125^{\circ}\). But this is incorrect because \(\angle2\) and \(\angle4\) are vertical angles, and \(\angle4\) and \(\angle6\) are alternate - interior angles (\(e\parallel f\)). \(\angle5 = 55^{\circ}\), \(\angle5+\angle6 = 180^{\circ}\) (linear pair), \(m\angle6 = 125^{\circ}\), so \(m\angle4 = 125^{\circ}\), \(m\angle2 = 125^{\circ}\) is wrong.
• For \(m\angle3\): As \(\angle3\) and \(\angle5\) are alternate - interior angles (\(e\parallel f\)), \(m\angle3 = 55^{\circ}\).
• For \(m\angle8\): \(\angle8\) and \(\angle3\) are alternate - exterior angles (\(e\parallel f\)), \(m\angle8 = 55^{\circ}\).
• For \(m\angle12\): \(\angle12\) and \(\angle10\) are vertical angles, \(\angle10\) and \(\angle9\) form a linear pair (\(\angle9 = 80^{\circ}\)), \(m\angle10=100^{\circ}\), so \(m\angle12 = 100^{\circ}\).
• For \(m\angle14\): \(\angle14\) and \(\angle12\) are vertical angles, \(m\angle14 = 100^{\circ}\) is wrong because \(\angle14\) and \(\angle16\) form a linear pair, \(\angle13\) and \(\angle11\) are vertical angles (\(m\angle11 = 80^{\circ}\)), \(\angle13+\angle14 = 180^{\circ}\), \(m\angle14=100^{\circ}\) is wrong.

Answer:

B. \(m\angle3 = 55^{\circ}\), C. \(m\angle8 = 55^{\circ}\), D. \(m\angle12 = 100^{\circ}\)