Question

two parallel lines are crossed by a transversal. if $m\angle 6 = 123.5^\circ$, then $m\angle 1$ is \bigcirc $56.5^\circ$. \bigcirc $67.5^\circ$. \bigcirc $123.5^\circ$. \bigcirc $136.5^\circ$. (there is a diagram of two parallel lines cut by a transversal with angles labeled 1,2,3,4,5,6,7,8)

Explanation:

Step1: Identify angle relationships

When two parallel lines are cut by a transversal, consecutive interior angles are supplementary (sum to \(180^\circ\)), and corresponding angles are equal. Also, vertical angles are equal. First, note that \(\angle 6\) and \(\angle 3\) are corresponding angles? Wait, no, let's look at the diagram. \(\angle 6\) and \(\angle 5\) are adjacent, but \(\angle 1\) and \(\angle 3\) are vertical angles? Wait, no, let's see: \(\angle 6\) and \(\angle 3\) – actually, \(\angle 6\) and \(\angle 1\): since the lines are parallel, \(\angle 6\) and \(\angle 3\) are same - side interior angles? Wait, no, let's correct. The two parallel lines are cut by a transversal. So, \(\angle 6\) and \(\angle 3\) are same - side interior angles? Wait, no, let's look at the positions. \(\angle 6\) and \(\angle 1\): \(\angle 6\) and \(\angle 5\) are supplementary (linear pair), \(\angle 5\) and \(\angle 1\) are corresponding angles (since the two lines are parallel). So first, find \(\angle 5\): since \(\angle 6\) and \(\angle 5\) are a linear pair, \(m\angle 5=180^\circ - m\angle 6\). Then, \(\angle 5\) and \(\angle 1\) are corresponding angles, so \(m\angle 1 = m\angle 5\).

Step2: Calculate \(m\angle 5\)

Given \(m\angle 6 = 123.5^\circ\), since \(\angle 5\) and \(\angle 6\) form a linear pair (they are adjacent and form a straight line), their measures add up to \(180^\circ\). So, \(m\angle 5=180^\circ - 123.5^\circ\)
\[m\angle 5 = 56.5^\circ\] Wait, no, that can't be. Wait, maybe I mixed up the angles. Wait, no, let's look again. Wait, \(\angle 1\) and \(\angle 6\): actually, \(\angle 1\) and \(\angle 6\) – no, let's see the diagram. The angle \(\angle 1\) and \(\angle 6\): the two parallel lines (the ones with the orange arrows) are cut by the transversal (line \(w\) and the other transversal). Wait, maybe \(\angle 1\) and \(\angle 6\) are not the right pair. Wait, let's look at vertical angles or corresponding angles. Wait, \(\angle 6\) and \(\angle 3\) are same - side interior angles? No, same - side interior angles are supplementary. Wait, \(\angle 6\) and \(\angle 3\): if the two parallel lines are the ones with the orange arrows, then the transversal is the line with angles 1,2,3,4. So \(\angle 6\) and \(\angle 3\) are same - side interior angles, so they should be supplementary. Then \(\angle 3\) and \(\angle 1\) are vertical angles, so \(m\angle 1=m\angle 3\). So first, find \(m\angle 3\): since \(\angle 6\) and \(\angle 3\) are supplementary, \(m\angle 3 = 180^\circ - m\angle 6\). Then \(m\angle 1=m\angle 3\) (vertical angles).

Wait, let's start over. The two parallel lines (let's say line \(x\) and line \(y\)) are cut by a transversal (the line with angles 1,2,3,4). So angle 6 is on the transversal cutting line \(y\) and the other parallel line. So angle 6 and angle 3: when two parallel lines are cut by a transversal, same - side interior angles are supplementary. So \(\angle 3\) and \(\angle 6\) are same - side interior angles, so \(m\angle 3 + m\angle 6=180^\circ\). Then, \(\angle 1\) and \(\angle 3\) are vertical angles, so \(m\angle 1 = m\angle 3\).

Step3: Calculate \(m\angle 3\)

Given \(m\angle 6 = 123.5^\circ\), then \(m\angle 3=180^\circ - 123.5^\circ=56.5^\circ\)? Wait, no, that's not matching. Wait, maybe I got the angle positions wrong. Wait, the diagram: angles 1,2,3,4 are on one transversal, angles 5,6,7,8 on the other. The two parallel lines are the ones with the orange arrows (line \(x\) and line \(y\)). So line \(x\) and line \(y\) are parallel, cut by the transversal (the line with angles 1 - 4). So angle 1 and angle 5: are they corresponding angles? Angle 1 is on line \(x\), angle 5 is on line \(y\), same position relative to the transversal. So \(\angle 1\) and \(\angle 5\) are corresponding angles, so they are equal. Angle 5 and angle 6 are a linear pair, so \(m\angle 5 + m\angle 6=180^\circ\). So \(m\angle 5 = 180^\circ - 123.5^\circ = 56.5^\circ\), and since \(\angle 1\) and \(\angle 5\) are corresponding angles (because the lines are parallel), \(m\angle 1=m\angle 5 = 56.5^\circ\).

Answer:

\(56.5^\circ\) (corresponding to the first option: 56.5°)