Question

1. if ( mangle3 = 153^{circ} ), then what are the measures of the remaining angles?

( mangle1 = 153^{circ}, mangle2 = 153^{circ}, mangle4 = 153^{circ} )
( mangle1 = 27^{circ}, mangle2 = 153^{circ}, mangle4 = 153^{circ} )
( mangle1 = 27^{circ}, mangle2 = 27^{circ}, mangle4 = 153^{circ} )
( mangle1 = 153^{circ}, mangle2 = 27^{circ}, mangle4 = 27^{circ} )

Explanation:

Step1: Vertical angles

Vertical angles are equal. Since \( \angle1 \) and \( \angle3 \) are vertical angles, \( m\angle1=m\angle3 = 153^{\circ} \).

Step2: Linear - pair angles

Angles in a linear pair are supplementary (\(m\angle2 + m\angle3=180^{\circ}\)). Given \(m\angle3 = 153^{\circ}\), then \(m\angle2=180^{\circ}-153^{\circ}=27^{\circ}\).

Step3: Vertical angles again

\( \angle2 \) and \( \angle4 \) are vertical angles. So \(m\angle4=m\angle2 = 27^{\circ}\).

Answer:

\(m\angle1 = 153^{\circ},m\angle2 = 27^{\circ},m\angle4 = 27^{\circ}\) (the fourth option)