Question

determining angle measures
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 the linear - pair relationship

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

Step2: Use the vertical - angle relationship

\(\angle12\) and \(\angle14\) are vertical angles. So \(m\angle14 = m\angle12 = 80^{\circ}\). \(\angle12\) and \(\angle11\) form a linear pair (\(m\angle11=180^{\circ}-m\angle12 = 100^{\circ}\)). \(\angle11\) and \(\angle13\) are vertical angles, so \(m\angle13=m\angle11 = 100^{\circ}\).

Step3: Use the corresponding - angle and alternate - interior - angle relationships

Since the two horizontal lines are parallel. \(\angle3\) and \(\angle5\) are alternate - interior angles, so \(m\angle5=m\angle3 = 120^{\circ}\). \(\angle3\) and \(\angle6\) form a linear pair (\(m\angle6=180^{\circ}-m\angle3=60^{\circ}\)). \(\angle10\) and \(\angle12\) form a linear pair (\(m\angle10 = 180^{\circ}-m\angle12=100^{\circ}\)).

Answer:

\(m\angle1 = 60^{\circ}\) is correct.
\(m\angle13 = 80^{\circ}\) is incorrect.
\(m\angle6 = 80^{\circ}\) is incorrect.
\(m\angle5 = 60^{\circ}\) is incorrect.
\(m\angle10 = 120^{\circ}\) is incorrect.
\(m\angle14 = 100^{\circ}\) is incorrect.
So the only correct one is \(m\angle1 = 60^{\circ}\), i.e., the option \(m\angle1 = 60^{\circ}\) is correct.