Question

in the figure, m is parallel to n and ( mangle4 = 122^{circ} ). find the measures of the other angles.
( mangle6 = 122^{circ} )
( mangle2 = 122^{circ} )
( mangle8 = square^{circ} )

Explanation:

Step1: Vertical angles

Vertical angles are equal. Since \(m\angle4 = 122^{\circ}\), and \(\angle4\) and \(\angle2\) are vertical angles, so \(m\angle2=122^{\circ}\).

Step2: Corresponding angles

\(m\) is parallel to \(n\). \(\angle6\) and \(\angle4\) are corresponding angles. Corresponding angles are equal when two parallel lines are cut by a transversal. So \(m\angle6 = 122^{\circ}\).

Step3: Linear - pair relationship

\(\angle8\) and \(\angle2\) form a linear pair. The sum of angles in a linear pair is \(180^{\circ}\). Let \(m\angle8=x\). Then \(x + 122^{\circ}=180^{\circ}\). Solving for \(x\), we get \(x=180^{\circ}-122^{\circ}=58^{\circ}\).

Answer:

\(m\angle2 = 122^{\circ}\), \(m\angle6=122^{\circ}\), \(m\angle8 = 58^{\circ}\)