Question

select all the angles that are supplementary to \\( \angle 1 \\).
\\( \square \angle 3 \\)
\\( \square \angle 4 \\)
\\( \square \angle 5 \\)
\\( \square \angle 6 \\)
\\( \square \angle 7 \\)
\\( \square \angle 8 \\)

Explanation:

Step1: Recall the definition of supplementary angles

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

Step2: Analyze the relationship between \(\angle1\) and other angles

• \(\angle1\) and \(\angle2\) form a linear pair, so \(\angle1+\angle2 = 180^{\circ}\). Also, \(\angle1\) and \(\angle3\) form a linear pair, so \(\angle1+\angle3=180^{\circ}\).
• Since \(l\parallel m\), by the property of parallel lines and transversal \(t\), \(\angle1\cong\angle5\) (corresponding angles). And \(\angle5+\angle6 = 180^{\circ}\) (linear pair), \(\angle5+\angle8=180^{\circ}\) (vertical - angle and linear - pair relationships: \(\angle7\cong\angle5\), \(\angle7+\angle8 = 180^{\circ}\)). Substituting \(\angle1\) for \(\angle5\), we get \(\angle1+\angle6=180^{\circ}\) and \(\angle1+\angle8 = 180^{\circ}\).

Answer:

\(\angle3\), \(\angle6\), \(\angle8\)