Question

two parallel lines are cut by a transversal as shown below. suppose m ∠6 = 148°. find m ∠1 and m ∠4.

Explanation:

Step1: Find \( m\angle 5 \) (supplementary to \( \angle 6 \))

Since \( \angle 5 \) and \( \angle 6 \) are supplementary (linear pair), \( m\angle 5 + m\angle 6 = 180^\circ \). Given \( m\angle 6 = 148^\circ \), we have \( m\angle 5 = 180^\circ - 148^\circ = 32^\circ \).

Step2: Find \( m\angle 1 \) (corresponding angles)

\( \angle 1 \) and \( \angle 5 \) are corresponding angles (since the lines are parallel and cut by a transversal). Corresponding angles are equal, so \( m\angle 1 = m\angle 5 = 32^\circ \).

Step3: Find \( m\angle 4 \) (same - side interior angles or supplementary to \( \angle 3 \), but easier via \( \angle 1 \))

\( \angle 1 \) and \( \angle 4 \) are vertical angles? No, wait, \( \angle 1 \) and \( \angle 4 \): Wait, actually, \( \angle 4 \) and \( \angle 5 \) are also corresponding? Wait, no. Wait, \( \angle 4 \) and \( \angle 6 \): Wait, let's re - examine. The two parallel lines: the upper parallel line, middle parallel line? Wait, the transversal cuts two parallel lines. \( \angle 4 \) and \( \angle 6 \): Wait, \( \angle 4 \) and \( \angle 5 \) are alternate interior angles? Wait, no, let's use the fact that \( \angle 1 \) and \( \angle 4 \): Wait, \( \angle 1 \) and \( \angle 3 \) are vertical angles, \( \angle 3 \) and \( \angle 5 \) are alternate interior angles (so \( \angle 3=\angle 5 = 32^\circ \)), and \( \angle 4 \) and \( \angle 3 \) are supplementary? Wait, no, \( \angle 4 \) and \( \angle 3 \) are adjacent angles forming a linear pair? Wait, no, the upper line: angles 1,2 on the top, 4,3 on the middle. The transversal goes through the middle line and the bottom line, creating 5,6,8,7. So \( \angle 4 \) and \( \angle 2 \) are supplementary, but \( \angle 2 \) and \( \angle 6 \) are corresponding angles. Wait, maybe a better way: \( \angle 1 \) and \( \angle 4 \): Since \( \angle 1 \) and \( \angle 4 \) are vertical angles? No, \( \angle 1 \) and \( \angle 3 \) are vertical angles. Wait, I made a mistake earlier. Let's correct:

Wait, \( \angle 1 \) and \( \angle 3 \) are vertical angles, so \( m\angle 1=m\angle 3 \). \( \angle 3 \) and \( \angle 5 \) are alternate interior angles (so \( m\angle 3 = m\angle 5=32^\circ \)), so \( m\angle 1 = 32^\circ \). Now, \( \angle 4 \) and \( \angle 1 \): Wait, \( \angle 4 \) and \( \angle 2 \) are supplementary, \( \angle 2 \) and \( \angle 6 \) are corresponding angles (so \( m\angle 2=m\angle 6 = 148^\circ \)), so \( m\angle 4=180^\circ - m\angle 2=180 - 148 = 32^\circ \)? No, that can't be. Wait, no, I messed up the parallel lines. Let's look at the diagram again. The two parallel lines: the bottom parallel line has angles 5,6,8,7. The middle parallel line has angles 4,3. The top parallel line has angles 1,2. The transversal goes from top left to bottom right. So \( \angle 1 \) and \( \angle 5 \): corresponding angles (since they are in the same position relative to the parallel lines and the transversal). So \( m\angle 1 = m\angle 5=32^\circ \). \( \angle 4 \) and \( \angle 1 \): Wait, \( \angle 4 \) and \( \angle 1 \): are they corresponding? Wait, no, \( \angle 4 \) and \( \angle 5 \): alternate interior angles? Wait, the middle and bottom parallel lines: transversal cuts them, so \( \angle 4 \) and \( \angle 5 \) are alternate interior angles, so \( m\angle 4=m\angle 5 = 32^\circ \)? Wait, no, that would mean \( \angle 4 = 32^\circ \), but let's check with \( \angle 6 \). \( \angle 4 \) and \( \angle 6 \): same - side interior angles? \( m\angle 4 + m\angle 6=32 + 148 = 180^\circ \), so they are supplementary. So \( m\angle 4 = 180^\circ - 148^\circ=32^\circ \)? Wait, no, that's the same as \( m\angle 1 \). Wait, maybe \( \angle 1 \) and \( \angle 4 \) are equal because of the parallel lines.

Wait, let's start over.

1. Relationship between \( \angle 6 \) and \( \angle 5 \): linear pair, so \( m\angle 5=180 - 148 = 32^\circ \).

2. \( \angle 1 \) and \( \angle 5 \): corresponding angles (parallel lines, transversal), so \( m\angle 1 = m\angle 5 = 32^\circ \).

3. \( \angle 4 \) and \( \angle 1 \): Wait, \( \angle 4 \) and \( \angle 1 \): are they vertical angles? No, \( \angle 1 \) and \( \angle 3 \) are vertical angles. \( \angle 3 \) and \( \angle 5 \) are alternate interior angles (so \( m\angle 3 = 32^\circ \)). \( \angle 4 \) and \( \angle 3 \) are supplementary (linear pair on the middle parallel line), so \( m\angle 4=180 - m\angle 3 = 180 - 32 = 148^\circ \)? Wait, now I'm confused. Wait, no, the middle parallel line: angles 4 and 3 are adjacent, forming a linear pair? So \( m\angle 4 + m\angle 3=180^\circ \). \( \angle 3 \) and \( \angle 5 \) are alternate interior angles (so \( m\angle 3 = m\angle 5 = 32^\circ \)), so \( m\angle 4=180 - 32 = 148^\circ \). And \( \angle 1 \) and \( \angle 3 \) are vertical angles, so \( m\angle 1 = m\angle 3 = 32^\circ \). Ah! I see my mistake earlier. So let's correct:

Step2 (corrected): Find \( m\angle 1 \) (vertical angles with \( \angle 3 \), alternate interior with \( \angle 5 \))

\( \angle 3 \) and \( \angle 5 \) are alternate interior angles (parallel lines, transversal), so \( m\angle 3=m\angle 5 \). \( \angle 5 \) and \( \angle 6 \) are supplementary, so \( m\angle 5 = 180 - 148=32^\circ \), so \( m\angle 3 = 32^\circ \). \( \angle 1 \) and \( \angle 3 \) are vertical angles, so \( m\angle 1=m\angle 3 = 32^\circ \).

Step3 (corrected): Find \( m\angle 4 \) (supplementary to \( \angle 3 \))

\( \angle 4 \) and \( \angle 3 \) form a linear pair (they are adjacent angles on a straight line), so \( m\angle 4 + m\angle 3=180^\circ \). Since \( m\angle 3 = 32^\circ \), then \( m\angle 4=180 - 32 = 148^\circ \).

Answer:

\( m\angle 1=\boxed{32}^\circ \)
\( m\angle 4=\boxed{148}^\circ \)