Question

two parallel lines are cut by a transversal as shown below. suppose $m\angle 8 = 37^\circ$. find $m\angle 2$ and $m\angle 3$. image of two parallel lines cut by a transversal with angles labeled 1,2,3,4 on the top line and 5,6,7,8 on the bottom line $m\angle 2 = \square^\circ$ $m\angle 3 = \square^\circ$

Explanation:

Step1: Find \( m\angle 2 \)

\(\angle 8\) and \(\angle 2\) are corresponding angles (since two parallel lines cut by a transversal, corresponding angles are equal). Given \( m\angle 8 = 37^\circ \), so \( m\angle 2 = 37^\circ \)? Wait, no, wait. Wait, \(\angle 8\) and \(\angle 2\): Wait, actually, \(\angle 8\) and \(\angle 2\) – wait, maybe I made a mistake. Wait, \(\angle 8\) and \(\angle 4\) are corresponding? No, wait, let's look at the diagram. The two parallel lines: the upper line has angles 1,2,4,3; the lower line has 5,6,8,7. So transversal crosses them. So \(\angle 8\) and \(\angle 4\) – no, \(\angle 8\) and \(\angle 2\): Wait, \(\angle 8\) and \(\angle 2\) – actually, \(\angle 8\) and \(\angle 2\) are not corresponding. Wait, \(\angle 8\) and \(\angle 4\) – no, wait, \(\angle 8\) and \(\angle 2\): Wait, maybe \(\angle 8\) and \(\angle 2\) are alternate exterior? No, wait, \(\angle 8\) is at the bottom left, \(\angle 2\) is at the top right. Wait, maybe \(\angle 8\) and \(\angle 2\) are vertical angles? No, \(\angle 8\) and \(\angle 6\) are vertical? Wait, no, \(\angle 8\) and \(\angle 7\) are adjacent supplementary, \(\angle 8\) and \(\angle 5\) are vertical? Wait, maybe I messed up. Wait, let's recall: when two parallel lines are cut by a transversal, alternate interior angles are equal, corresponding angles are equal, consecutive interior angles are supplementary. Also, vertical angles are equal, linear pairs are supplementary.

Wait, \(\angle 8\) and \(\angle 2\): Let's see, \(\angle 8\) and \(\angle 2\) – are they corresponding? Wait, the transversal: the upper line's angle 2 and lower line's angle 8 – no, maybe \(\angle 8\) and \(\angle 2\) are not. Wait, maybe \(\angle 8\) and \(\angle 4\) are corresponding? No, \(\angle 4\) is on the upper line, left side. \(\angle 8\) is on the lower line, left side. So \(\angle 4\) and \(\angle 8\) are corresponding, so \( m\angle 4 = 37^\circ \). Then \(\angle 2\) and \(\angle 4\) are vertical angles? No, \(\angle 2\) and \(\angle 4\) are adjacent? Wait, no, \(\angle 1\) and \(\angle 2\) are linear pair, \(\angle 1\) and \(\angle 4\) are vertical? Wait, maybe I need to re-examine.

Wait, let's correct: \(\angle 8\) and \(\angle 2\) – actually, \(\angle 8\) and \(\angle 2\) are equal? Wait, no, maybe \(\angle 8\) and \(\angle 2\) are alternate exterior angles? Wait, alternate exterior angles: when two parallel lines are cut by a transversal, alternate exterior angles are equal. So \(\angle 8\) is an exterior angle (below the lower line, left side), \(\angle 2\) is an exterior angle (above the upper line, right side). So they are alternate exterior angles, so they should be equal. So \( m\angle 2 = m\angle 8 = 37^\circ \)? Wait, but then \(\angle 3\) and \(\angle 2\) are linear pair? Wait, \(\angle 2\) and \(\angle 3\) are adjacent, forming a linear pair, so they are supplementary. So \( m\angle 3 = 180^\circ - m\angle 2 = 180^\circ - 37^\circ = 143^\circ \). Wait, but that seems off. Wait, maybe I made a mistake in identifying the angles.

Wait, let's start over. Let's label the angles:

- Upper line: angles 1 (top left), 2 (top right), 4 (bottom left), 3 (bottom right)
- Lower line: angles 5 (top left), 6 (top right), 8 (bottom left), 7 (bottom right)

Transversal crosses both lines. So:

- \(\angle 8\) and \(\angle 4\): corresponding angles? No, \(\angle 8\) is on lower line, bottom left; \(\angle 4\) is on upper line, bottom left. So they are corresponding angles. So \( m\angle 4 = m\angle 8 = 37^\circ \).

- \(\angle 4\) and \(\angle 2\): vertical angles? No, \(\angle 1\) and \(\angle 3\) are vertical, \(\angle 2\) and \(\angle 4\) are vertical? Wait, \(\angle 1\) and \(\angle 3\) are vertical (opposite each other), \(\angle 2\) and \(\angle 4\) are vertical? Wait, no, \(\angle 1\) and \(\angle 3\) are vertical, \(\angle 2\) and \(\angle 4\) are vertical? Wait, the intersection of the transversal and the upper line: angle 1 (top left), angle 2 (top right), angle 4 (bottom left), angle 3 (bottom right). So angle 1 and angle 3 are vertical, angle 2 and angle 4 are vertical. So \( m\angle 2 = m\angle 4 = 37^\circ \). Then angle 2 and angle 3 are adjacent, forming a linear pair, so they are supplementary. So \( m\angle 3 = 180^\circ - 37^\circ = 143^\circ \).

Yes, that makes sense. So:

Step1: Find \( m\angle 2 \)

\(\angle 8\) and \(\angle 4\) are corresponding angles (parallel lines cut by transversal), so \( m\angle 4 = m\angle 8 = 37^\circ \). \(\angle 2\) and \(\angle 4\) are vertical angles, so \( m\angle 2 = m\angle 4 = 37^\circ \).

Step2: Find \( m\angle 3 \)

\(\angle 2\) and \(\angle 3\) form a linear pair (supplementary), so \( m\angle 3 = 180^\circ - m\angle 2 = 180^\circ - 37^\circ = 143^\circ \).

Answer:

\( m\angle 2 = \boxed{37}^\circ \)
\( m\angle 3 = \boxed{143}^\circ \)