Question

in the figure shown, ( ell_1 ) and ( ell_2 ) are not parallel. which angles are supplements of ( <1 )? a. angles 6, 3, and 8. b. angles 2 and 5. c. angles 2, 5, 4, and 7. d. angles 5 and 7.

Explanation:

Step1: Recall the definition of supplementary angles

Supplementary angles are two angles whose sum is \(180^{\circ}\).

Step2: Analyze angle - angle relationships

• Angle 2: \(\angle1+\angle2 = 180^{\circ}\) (linear - pair postulate, they form a straight line).
• Angle 5: \(\angle1\) and \(\angle5\) are vertical angles to \(\angle2\) and \(\angle6\) respectively. Since \(\angle2+\angle6=180^{\circ}\) (linear pair) and \(\angle1=\angle6\), \(\angle5=\angle2\), then \(\angle1 + \angle5=180^{\circ}\).
• Angle 4: \(\angle1\) and \(\angle4\) are related through the transversal and non - parallel lines. \(\angle1\) and \(\angle6\) are vertical angles (\(\angle1=\angle6\)), \(\angle6+\angle7 = 180^{\circ}\) (linear pair), \(\angle7=\angle4\) (vertical angles). So \(\angle1+\angle4=180^{\circ}\).
• Angle 7: \(\angle1\) and \(\angle6\) are vertical angles (\(\angle1=\angle6\)), \(\angle6+\angle7 = 180^{\circ}\) (linear pair). So \(\angle1+\angle7=180^{\circ}\).

Answer:

C. Angles 2, 5, 4, and 7.