Question

lines ( l_1 ) and ( l_2 ) are parallel lines. determine the measures of ( angle 1 ) through ( angle 12 ). ( mangle 7 = 50^{circ} ). ( mangle 1 = )

Explanation:

Step1: Use vertical angles property

Vertical angles are equal. So, \(m\angle1 = 64^{\circ}\) (since \(\angle1\) and the \(64^{\circ}\) angle are vertical angles).

Step2: Use linear - pair property

\(\angle1\) and \(\angle2\) form a linear pair. So, \(m\angle1+m\angle2 = 180^{\circ}\). Substituting \(m\angle1 = 64^{\circ}\), we get \(64^{\circ}+m\angle2 = 180^{\circ}\), then \(m\angle2=180^{\circ}- 64^{\circ}=116^{\circ}\).

Step3: Use vertical angles property

\(\angle2\) and \(\angle3\) are vertical angles. So, \(m\angle3 = m\angle2 = 116^{\circ}\).

Step4: Use vertical angles property

\(\angle4\) and the \(64^{\circ}\) angle form a linear pair. \(m\angle4 = 180^{\circ}-64^{\circ}=116^{\circ}\).

Step5: Use vertical angles property

\(\angle4\) and \(\angle5\) are vertical angles. So, \(m\angle5 = m\angle4 = 116^{\circ}\).

Step6: Use alternate - interior angles property (since \(L_1\parallel L_2\))

The angle adjacent to \(\angle6\) (which is \(66^{\circ}\)) and \(\angle7\) are alternate - interior angles. \(\angle6\) and the \(66^{\circ}\) angle are vertical angles, so \(m\angle6 = 66^{\circ}\). Then, using the linear - pair property for the \(66^{\circ}\) angle and the angle adjacent to \(\angle6\), we know that the angle adjacent to \(\angle6\) is \(180^{\circ}-66^{\circ}=114^{\circ}\). Since \(L_1\parallel L_2\), \(m\angle7 = 66^{\circ}\) (alternate - interior angles).

Step7: Use vertical angles property

\(\angle7\) and \(\angle8\) are vertical angles. So, \(m\angle8 = m\angle7 = 66^{\circ}\).

Step8: Use vertical angles property

\(\angle9\) and the \(66^{\circ}\) angle are vertical angles. So, \(m\angle9 = 66^{\circ}\).

Step9: Use linear - pair property

\(\angle9\) and \(\angle10\) form a linear pair. \(m\angle10=180^{\circ}-66^{\circ}=114^{\circ}\).

Step10: Use vertical angles property

\(\angle10\) and \(\angle11\) are vertical angles. So, \(m\angle11 = m\angle10 = 114^{\circ}\).

Step11: Use vertical angles property

\(\angle11\) and \(\angle12\) are vertical angles. So, \(m\angle12 = m\angle11 = 114^{\circ}\).

Answer:

\(m\angle1 = 64^{\circ}\), \(m\angle2 = 116^{\circ}\), \(m\angle3 = 116^{\circ}\), \(m\angle4 = 116^{\circ}\), \(m\angle5 = 116^{\circ}\), \(m\angle6 = 66^{\circ}\), \(m\angle7 = 66^{\circ}\), \(m\angle8 = 66^{\circ}\), \(m\angle9 = 66^{\circ}\), \(m\angle10 = 114^{\circ}\), \(m\angle11 = 114^{\circ}\), \(m\angle12 = 114^{\circ}\)