Question

in the diagram, ( mangle3 = 120^{circ} ) and ( mangle12 = 80^{circ} ). which angle measures are correct? check all that apply.
( square mangle1 = 60^{circ} )
( square mangle13 = 80^{circ} )
( square mangle6 = 80^{circ} )
( square mangle5 = 60^{circ} )
( square mangle10 = 120^{circ} )
( square mangle14 = 100^{circ} )

Explanation:

Step1: Use linear - pair relationship

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

Step2: Use vertical - angle relationship

\(\angle12\) and \(\angle13\) are vertical angles. By the vertical - angles theorem (\(\angle12\cong\angle13\)), so \(m\angle13 = m\angle12=80^{\circ}\).

Step3: Use alternate - interior angles and linear - pair relationships

\(\angle3\) and \(\angle5\) are alternate - interior angles (assuming the parallel lines cut by a transversal). But also, \(\angle3 = 120^{\circ}\), and \(\angle5\) and \(\angle3\) are related as \(\angle5=180 - 120 = 60^{\circ}\) (linear - pair with \(\angle7\) which is congruent to \(\angle3\) if we consider parallel lines). \(\angle6\) and \(\angle12\) are not congruent. \(\angle10\) and \(\angle12\) form a linear pair (\(m\angle10=180 - 80=100^{\circ}\)). \(\angle14\) and \(\angle12\) form a linear pair (\(m\angle14=180 - 80 = 100^{\circ}\)).

Answer:

\(m\angle1 = 60^{\circ}\), \(m\angle13 = 80^{\circ}\), \(m\angle5 = 60^{\circ}\)