Question

which of the angles in the figure are supplementary to ∠3?
figure with lines m (horizontal), n (horizontal, parallel to m), and a transversal, forming angles 1, 2, 3, 4 at the intersection with m; angles 5, 6, 7, 8 at the intersection with n.
options:
○ ∠2 and ∠4
○ ∠2
○ ∠1, ∠4, ∠5 and ∠8
○ ∠2, ∠7, and ∠6

Explanation:

Step1: Recall Supplementary Angles

Supplementary angles sum to \(180^\circ\). Vertical angles are equal, and linear pairs are supplementary. Also, corresponding angles (if lines are parallel) are equal, and alternate interior/exterior angles are equal.

Step2: Analyze \(\angle 3\)

• \(\angle 3\) and \(\angle 1\): Linear pair, so \(\angle 3 + \angle 1 = 180^\circ\) (supplementary).
• \(\angle 3\) and \(\angle 4\): Linear pair, so \(\angle 3 + \angle 4 = 180^\circ\) (supplementary).
• \(\angle 3\) and \(\angle 5\): If lines \(m\parallel n\), \(\angle 3\) and \(\angle 5\) are same - side interior angles? Wait, no, \(\angle 3\) and \(\angle 5\): Wait, \(\angle 3\) and \(\angle 5\) are actually, since \(\angle 3=\angle 2\) (vertical angles? No, \(\angle 3\) and \(\angle 2\) are adjacent? Wait, no, \(\angle 3\) and \(\angle 1\) are vertical? No, let's re - look. The transversal intersects line \(m\) at a point, creating \(\angle 1,\angle 2,\angle 3,\angle 4\). \(\angle 1\) and \(\angle 3\) are vertical? No, \(\angle 1\) and \(\angle 4\) are vertical, \(\angle 2\) and \(\angle 3\) are vertical? Wait, no, when two lines intersect, vertical angles are opposite. So at the intersection of the transversal and line \(m\), \(\angle 1\) and \(\angle 4\) are vertical, \(\angle 2\) and \(\angle 3\) are vertical? Wait, no, \(\angle 1+\angle 2 = 180^\circ\) (linear pair), \(\angle 2+\angle 4=180^\circ\), \(\angle 4 + \angle 3=180^\circ\), \(\angle 3+\angle 1 = 180^\circ\). Then, for the other intersection (transversal and line \(n\)), \(\angle 5\) and \(\angle 8\) are vertical, \(\angle 6\) and \(\angle 7\) are vertical. Also, if \(m\parallel n\), \(\angle 3=\angle 6\) (alternate interior), \(\angle 2=\angle 5\) (corresponding), \(\angle 1=\angle 8\) (corresponding), \(\angle 4=\angle 5\) (alternate interior)? Wait, maybe a better approach: supplementary to \(\angle 3\) are angles that add up to \(180^\circ\) with \(\angle 3\).

• \(\angle 1\): Linear pair with \(\angle 3\) (since they form a straight line on line \(m\)), so \(\angle 1+\angle 3 = 180^\circ\).
• \(\angle 4\): Linear pair with \(\angle 3\) (on line \(m\)), so \(\angle 4+\angle 3=180^\circ\).
• \(\angle 5\): If we consider the parallel lines (assuming \(m\parallel n\), which is implied by the angle relationships), \(\angle 3\) and \(\angle 5\): Wait, no, \(\angle 3\) and \(\angle 5\) are same - side interior angles? Wait, no, \(\angle 3\) and \(\angle 5\): Let's see, \(\angle 3\) and \(\angle 2\) are vertical (so \(\angle 3=\angle 2\)), and \(\angle 2\) and \(\angle 5\) are corresponding angles (if \(m\parallel n\)), so \(\angle 2=\angle 5\), so \(\angle 3=\angle 5\)? No, that can't be. Wait, maybe I made a mistake. Wait, the correct way: supplementary angles to \(\angle 3\) are angles that are adjacent (linear pair) or equal to angles that are supplementary to \(\angle 3\) via parallel lines.

Wait, \(\angle 3\) and \(\angle 1\): linear pair (supplementary). \(\angle 3\) and \(\angle 4\): linear pair (supplementary). Now, \(\angle 1\) and \(\angle 8\) are vertical (so \(\angle 1 = \angle 8\)), and \(\angle 4\) and \(\angle 5\) are alternate interior angles (if \(m\parallel n\)), so \(\angle 4=\angle 5\). So \(\angle 3+\angle 8=\angle 3+\angle 1 = 180^\circ\) (since \(\angle 1=\angle 8\)), and \(\angle 3+\angle 5=\angle 3+\angle 4 = 180^\circ\) (since \(\angle 4=\angle 5\)). So the angles supplementary to \(\angle 3\) are \(\angle 1\), \(\angle 4\), \(\angle 5\), and \(\angle 8\).

Answer:

\(\angle1,\angle4,\angle5\) and \(\angle8\) (the option with \(\angle1,\angle4,\angle5\) and \(\angle8\))