Question

in the figure below, ( mangle1 = 32^{circ} ). find ( mangle2 ), ( mangle3 ), and ( mangle4 ).

Explanation:

Step1: Use vertical angles property

Vertical angles are equal. So \(m\angle2 = m\angle1\) since \(\angle1\) and \(\angle2\) are vertical angles. Given \(m\angle1 = 32^{\circ}\), so \(m\angle2=32^{\circ}\).

Step2: Use linear - pair property

\(\angle1\) and \(\angle3\) form a linear pair. The sum of angles in a linear pair is \(180^{\circ}\). So \(m\angle1 + m\angle3=180^{\circ}\). Substitute \(m\angle1 = 32^{\circ}\), then \(32^{\circ}+m\angle3 = 180^{\circ}\). Solve for \(m\angle3\): \(m\angle3=180^{\circ}- 32^{\circ}=148^{\circ}\).

Step3: Use vertical angles property again

\(\angle3\) and \(\angle4\) are vertical angles. So \(m\angle4 = m\angle3\). Since \(m\angle3 = 148^{\circ}\), then \(m\angle4 = 148^{\circ}\).

Answer:

\(m\angle2 = 32^{\circ}\), \(m\angle3 = 148^{\circ}\), \(m\angle4 = 148^{\circ}\)