Question

in the figure below, ( mangle2 = 74^{circ} ). find ( mangle1 ), ( mangle3 ), and ( mangle4 ).

Explanation:

Step1: Use the property of vertical angles

Vertical angles are equal. So \(m\angle1 = m\angle3\) and \(m\angle2 = m\angle4\).

Step2: Use the property of supplementary angles

\(\angle1\) and \(\angle2\) are supplementary (they form a linear pair), so \(m\angle1+m\angle2 = 180^{\circ}\). Given \(m\angle2 = 74^{\circ}\), then \(m\angle1=180^{\circ}- 74^{\circ}=106^{\circ}\).

Step3: Find \(m\angle3\)

Since \(m\angle1 = m\angle3\) (vertical angles), \(m\angle3 = 106^{\circ}\).

Step4: Find \(m\angle4\)

Since \(m\angle2 = m\angle4\) (vertical angles), \(m\angle4 = 74^{\circ}\).

Answer:

\(m\angle1 = 106^{\circ}\), \(m\angle3 = 106^{\circ}\), \(m\angle4 = 74^{\circ}\)