Question

lines e and f are parallel. the ( mangle9 = 80^{circ} ) and ( mangle5 = 55^{circ} ).
which angle measures are correct? select three options.
( square mangle2 = 125^{circ} )
( square mangle3 = 55^{circ} )
( square mangle8 = 55^{circ} )
( square mangle12 = 100^{circ} )
( square mangle14 = 100^{circ} )

Explanation:

Step1: Use linear - pair and parallel - line properties

Since \(m\angle9 = 80^{\circ}\), and \(\angle9\) and \(\angle11\) are a linear pair (\(\angle9+\angle11 = 180^{\circ}\)), then \(m\angle11=180 - 80=100^{\circ}\). Also, \(\angle11\) and \(\angle12\) are a linear pair (\(\angle11+\angle12 = 180^{\circ}\)), but \(\angle9\) and \(\angle12\) are vertical angles. So \(m\angle12 = 80^{\circ}\).

Since \(m\angle5 = 55^{\circ}\), and \(\angle5\) and \(\angle3\) are alternate interior angles (because lines \(e\) and \(f\) are parallel), so \(m\angle3=m\angle5 = 55^{\circ}\).

\(\angle5\) and \(\angle8\) are vertical angles. By the vertical - angle theorem (\(\angle5=\angle8\)), so \(m\angle8 = 55^{\circ}\).

\(\angle2\) and \(\angle6\) are corresponding angles. First, find \(\angle6\). Since \(\angle5 = 55^{\circ}\) and \(\angle5+\angle6 = 180^{\circ}\) (linear pair), \(m\angle6=180 - 55=125^{\circ}\). So \(m\angle2=m\angle6 = 125^{\circ}\).

\(\angle14\) and \(\angle10\) are corresponding angles. Since \(\angle9 = 80^{\circ}\) and \(\angle9+\angle10 = 180^{\circ}\) (linear pair), \(m\angle10 = 100^{\circ}\), so \(m\angle14=m\angle10 = 100^{\circ}\).

Answer:

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