Question

in the figure below, ( mangle1 = 77^{circ} ). find ( mangle2 ), ( mangle3 ), and ( mangle4 ).

Explanation:

Step1: Use the property of supplementary angles

Supplementary angles sum to \(180^{\circ}\). Since \(\angle1\) and \(\angle2\) are supplementary, \(m\angle1 + m\angle2=180^{\circ}\). Given \(m\angle1 = 77^{\circ}\), then \(m\angle2=180^{\circ}-77^{\circ}\)

$$m\angle2 = 103^{\circ}$$

Step2: Use the property of vertical angles

Vertical angles are equal. \(\angle1\) and \(\angle3\) are vertical angles, so \(m\angle3=m\angle1\)

$$m\angle3 = 77^{\circ}$$

Step3: Use the property of vertical angles again

\(\angle2\) and \(\angle4\) are vertical angles. Since \(m\angle2 = 103^{\circ}\), then \(m\angle4=m\angle2\)

$$m\angle4 = 103^{\circ}$$

Answer:

\(m\angle2 = 103^{\circ}\), \(m\angle3 = 77^{\circ}\), \(m\angle4 = 103^{\circ}\)