Question

complete the statement.
if the measure of $\angle 6 = 120^\circ$, then the measure of $\angle 8 = $
\\(\circ\\) $50^\circ$
\\(\circ\\) $60^\circ$
\\(\circ\\) $120^\circ$
\\(\circ\\) $130^\circ$

Explanation:

Step1: Identify Angle Relationships

Lines \(a\) and \(b\) are parallel (indicated by arrows), and line \(c\) is a transversal. \(\angle 6\) and \(\angle 2\) are supplementary (linear pair), but \(\angle 8\) and \(\angle 6\) – wait, actually, \(\angle 6\) and \(\angle 2\) are supplementary, \(\angle 2\) and \(\angle 4\) are equal (alternate interior), \(\angle 4\) and \(\angle 8\) are vertical angles? No, wait, \(\angle 6\) and \(\angle 1\) are supplementary (linear pair), \(\angle 1\) and \(\angle 4\) – no, better: \(\angle 6\) and \(\angle 2\) are supplementary (\(180^\circ\)), \(\angle 2\) and \(\angle 4\) are equal (alternate interior), \(\angle 4\) and \(\angle 8\) are vertical angles? Wait, no, \(\angle 3\) and \(\angle 8\) – wait, actually, \(\angle 6\) and \(\angle 2\) are supplementary, \(\angle 2\) and \(\angle 4\) are equal (alternate interior), \(\angle 4\) and \(\angle 8\) are vertical angles? No, \(\angle 3\) and \(\angle 8\) are vertical? Wait, maybe simpler: \(\angle 6\) and \(\angle 2\) are supplementary (\(180^\circ\)), so \(\angle 2 = 180^\circ - 120^\circ = 60^\circ\). Then \(\angle 2\) and \(\angle 4\) are equal (alternate interior), so \(\angle 4 = 60^\circ\). Then \(\angle 4\) and \(\angle 8\) are vertical angles? No, \(\angle 3\) and \(\angle 8\) – wait, \(\angle 3\) and \(\angle 8\) are vertical? Wait, no, \(\angle 7\) and \(\angle 8\) are supplementary, \(\angle 7\) and \(\angle 3\) are vertical. Wait, maybe I messed up. Let's look at the diagram: lines \(b\) and \(a\) are parallel, transversal \(c\). \(\angle 6\) and \(\angle 2\) are adjacent supplementary (linear pair), so \(\angle 2 = 60^\circ\). Then \(\angle 2\) and \(\angle 4\) are alternate interior angles (since \(b \parallel a\), transversal \(c\)? Wait, no, transversal is the horizontal line. Wait, the horizontal line is \(c\), and lines \(b\) and \(a\) are the two slanted lines. So \(\angle 6\) is on line \(a\), \(\angle 2\) is between \(b\) and \(a\) on \(c\). Then \(\angle 2\) and \(\angle 4\) are alternate interior angles (since \(b \parallel a\), transversal \(c\)? No, transversal is the horizontal line. Wait, \(\angle 2\) and \(\angle 4\) are alternate interior angles (between \(b\) and \(a\), cut by transversal \(c\)), so they are equal. So \(\angle 4 = \angle 2 = 60^\circ\). Then \(\angle 4\) and \(\angle 8\) are vertical angles? No, \(\angle 3\) and \(\angle 8\) are vertical. Wait, \(\angle 3\) and \(\angle 4\) are supplementary? No, \(\angle 3\) and \(\angle 4\) are adjacent supplementary (linear pair), so \(\angle 3 + \angle 4 = 180^\circ\). Wait, I'm overcomplicating. Let's use corresponding angles or alternate interior. Since \(b \parallel a\), \(\angle 8\) and \(\angle 2\) – no, \(\angle 8\) and \(\angle 6\): wait, \(\angle 6\) and \(\angle 1\) are supplementary, \(\angle 1\) and \(\angle 4\) – no. Wait, the correct approach: \(\angle 6\) and \(\angle 2\) are supplementary (linear pair), so \(\angle 2 = 60^\circ\). Then \(\angle 2\) and \(\angle 4\) are alternate interior angles (since \(b \parallel a\), transversal is the horizontal line), so \(\angle 4 = 60^\circ\). Then \(\angle 4\) and \(\angle 8\) are vertical angles? No, \(\angle 3\) and \(\angle 8\) are vertical. Wait, \(\angle 3\) and \(\angle 7\) are vertical, \(\angle 4\) and \(\angle 8\) are vertical? Yes! Because \(\angle 4\) and \(\angle 8\) are opposite each other when two lines intersect (the slanted line \(b\) and horizontal line \(c\) intersect at the left, forming \(\angle 3, \angle 4, \angle 7, \angle 8\)). So \(\angle 4\) and \(\angle 8\) are vertical angles, so they are equal. Since \(\angle 4 = 60^\circ\), \(\angle 8 = 60^\circ\). Alternatively, \(\angle 6\) and \(\angle 8\) – wait, no, \(\angle 6\) is \(120^\circ\), \(\angle 2\) is \(60^\circ\), \(\angle 2\) and \(\angle 4\) are equal, \(\angle 4\) and \(\angle 8\) are equal, so \(\angle 8 = 60^\circ\).

Step2: Calculate \(\angle 8\)

Since \(\angle 6 = 120^\circ\), \(\angle 6\) and \(\angle 2\) are supplementary (linear pair), so \(\angle 2 = 180^\circ - 120^\circ = 60^\circ\).
Lines \(b\) and \(a\) are parallel, so \(\angle 2\) and \(\angle 4\) are alternate interior angles (equal), so \(\angle 4 = 60^\circ\).
\(\angle 4\) and \(\angle 8\) are vertical angles (equal), so \(\angle 8 = \angle 4 = 60^\circ\).

Answer:

\(60^\circ\) (corresponding to the option \(60^\circ\))