Question

determining angle measures
in the diagram, ( mangle3 = 120^{circ} ) and ( mangle12 = 80^{circ} ). which angle measures are
correct? choose four correct answers
( mangle14 = 100^{circ} )
( mangle6 = 80^{circ} )
( mangle5 = 60^{circ} )
( mangle1 = 60^{circ} )

Explanation:

Step1: Use linear - pair relationship

Since \(\angle3\) and \(\angle1\) form a linear pair (\(\angle1+\angle3 = 180^{\circ}\)), and \(m\angle3=120^{\circ}\), then \(m\angle1=180^{\circ}-m\angle3\).
\(m\angle1 = 180 - 120=60^{\circ}\)

Step2: Use vertical - angle relationship

\(\angle3\) and \(\angle5\) are vertical angles. Vertical angles are equal. So \(m\angle5=m\angle3 = 60^{\circ}\)

Step3: Use linear - pair relationship for \(\angle12\) and \(\angle14\)

Since \(\angle12\) and \(\angle14\) form a linear pair (\(\angle12+\angle14 = 180^{\circ}\)), and \(m\angle12 = 80^{\circ}\), then \(m\angle14=180^{\circ}-m\angle12\)
\(m\angle14=180 - 80=100^{\circ}\)

Step4: Use alternate - interior angles (if lines are parallel)

Assume lines \(e\) and \(f\) are parallel. \(\angle6\) and \(\angle12\) are not alternate - interior angles. \(\angle6\) and \(\angle8\) are vertical angles. \(\angle12\) and \(\angle14\) form a linear pair (\(m\angle14 = 100^{\circ}\)), \(\angle6\) and \(\angle5\) form a linear pair (\(m\angle5 = 60^{\circ}\)), so \(m\angle6=120^{\circ}
eq80^{\circ}\)

Answer:

\(m\angle14 = 100^{\circ}\), \(m\angle5 = 60^{\circ}\), \(m\angle1 = 60^{\circ}\)