Question

two parallel lines are cut by a transversal as shown below. suppose m∠5 = 31°. find m∠2 and m∠4. m∠2 = \boxed{\circ} m∠4 = \boxed{\circ}

Explanation:

Step1: Find \( m\angle 2 \)

\(\angle 2\) and \(\angle 5\) are same - side interior angles? No, wait, \(\angle 2\) and \(\angle 5\) are actually same - side interior angles? Wait, no, when two parallel lines are cut by a transversal, same - side interior angles are supplementary? Wait, no, \(\angle 2\) and \(\angle 5\): Let's look at the positions. Wait, actually, \(\angle 2\) and \(\angle 5\) are same - side interior angles? Wait, no, \(\angle 2\) and \(\angle 5\): the two parallel lines are horizontal, and the transversal is the slant line. \(\angle 2\) and \(\angle 5\) are same - side interior angles, so they are supplementary. Wait, no, wait, \(\angle 2\) and \(\angle 5\): Let's check the vertical angles and corresponding angles. Wait, \(\angle 2\) and \(\angle 5\): actually, \(\angle 2\) and \(\angle 5\) are same - side interior angles, so \(m\angle 2 + m\angle 5=180^{\circ}\)? Wait, no, that's not right. Wait, maybe I made a mistake. Wait, \(\angle 2\) and \(\angle 5\): let's see, \(\angle 2\) and \(\angle 3\) are vertical angles? No, \(\angle 2\) and \(\angle 4\) are vertical angles. Wait, \(\angle 5\) and \(\angle 3\) are alternate interior angles. So \(\angle 3=\angle 5 = 31^{\circ}\). Then \(\angle 2\) and \(\angle 3\) are supplementary (linear pair), so \(m\angle 2=180 - 31=149^{\circ}\). Wait, that makes sense. Because \(\angle 3\) and \(\angle 5\) are alternate interior angles (since the two lines are parallel), so \(m\angle 3 = m\angle 5=31^{\circ}\). Then \(\angle 2\) and \(\angle 3\) form a linear pair, so \(m\angle 2=180 - 31 = 149^{\circ}\).

Step2: Find \( m\angle 4 \)

\(\angle 4\) and \(\angle 2\) are vertical angles? Wait, no, \(\angle 4\) and \(\angle 2\): \(\angle 2\) and \(\angle 4\) are vertical angles? Wait, no, \(\angle 2\) and \(\angle 4\) are adjacent and form a linear pair? Wait, no, the intersection of the transversal and the top parallel line: \(\angle 1\), \(\angle 2\), \(\angle 3\), \(\angle 4\) are around the intersection. \(\angle 2\) and \(\angle 4\) are vertical angles? No, \(\angle 1\) and \(\angle 3\) are vertical angles, \(\angle 2\) and \(\angle 4\) are vertical angles? Wait, no, \(\angle 2\) and \(\angle 4\) are adjacent and supplementary? Wait, no, at the intersection of two lines, vertical angles are equal, and linear pairs are supplementary. Wait, \(\angle 3\) and \(\angle 4\) are linear pairs, \(\angle 2\) and \(\angle 3\) are linear pairs. Wait, \(\angle 4\) and \(\angle 2\) are vertical angles? No, \(\angle 2\) and \(\angle 4\): if \(\angle 3 = 31^{\circ}\), and \(\angle 4\) and \(\angle 3\) are linear pairs, so \(m\angle 4=180 - 31 = 149^{\circ}\)? Wait, no, that can't be. Wait, no, \(\angle 4\) and \(\angle 2\): let's see, \(\angle 4\) and \(\angle 2\) are vertical angles? Wait, no, \(\angle 1\) and \(\angle 3\) are vertical angles, \(\angle 2\) and \(\angle 4\) are vertical angles? Wait, no, when two lines intersect, the vertical angles are equal. So \(\angle 1=\angle 3\), \(\angle 2=\angle 4\). And \(\angle 3=\angle 5 = 31^{\circ}\) (alternate interior angles). So \(\angle 4=\angle 2\), and \(\angle 2 = 180 - 31=149^{\circ}\), so \(\angle 4 = 149^{\circ}\)? Wait, no, wait, \(\angle 4\) and \(\angle 5\): \(\angle 4\) and \(\angle 5\): \(\angle 4\) and \(\angle 5\) are same - side interior angles? No, \(\angle 4\) and \(\angle 5\): let's check the positions. The top parallel line, the transversal, and the bottom parallel line. \(\angle 4\) and \(\angle 5\): \(\angle 4\) is above the top parallel line, \(\angle 5\) is below the top parallel line and above the bottom parallel line. Wait, maybe I messed up the alternate interior angles. Let's start over.

Two parallel lines (let's call them \(l_1\) (top) and \(l_2\) (bottom)) cut by a transversal \(t\).

- Alternate interior angles: \(\angle 3\) and \(\angle 5\) (since \(l_1\parallel l_2\)), so \(m\angle 3=m\angle 5 = 31^{\circ}\).
- Linear pair: \(\angle 2\) and \(\angle 3\) form a linear pair, so \(m\angle 2 + m\angle 3=180^{\circ}\). So \(m\angle 2=180 - 31 = 149^{\circ}\).
- Vertical angles: \(\angle 4\) and \(\angle 2\) are vertical angles? Wait, no, \(\angle 4\) and \(\angle 2\): at the intersection of \(l_1\) and \(t\), \(\angle 2\) and \(\angle 4\) are vertical angles? Wait, no, \(\angle 1\) and \(\angle 3\) are vertical angles, \(\angle 2\) and \(\angle 4\) are vertical angles. So \(m\angle 4=m\angle 2 = 149^{\circ}\)? Wait, no, that can't be. Wait, no, \(\angle 4\) and \(\angle 3\) are linear pairs? Wait, \(\angle 3\) and \(\angle 4\) are adjacent, so \(m\angle 3 + m\angle 4=180^{\circ}\), so \(m\angle 4 = 180 - 31=149^{\circ}\), which is the same as \(m\angle 2\) because \(\angle 2\) and \(\angle 4\) are vertical angles? Wait, yes, vertical angles are equal, and \(\angle 2\) and \(\angle 4\) are vertical angles, so \(m\angle 4=m\angle 2 = 149^{\circ}\). Wait, but let's check again. \(\angle 5 = 31^{\circ}\), \(\angle 3=\angle 5 = 31^{\circ}\) (alternate interior angles). \(\angle 2\) and \(\angle 3\) are supplementary (linear pair), so \(m\angle 2 = 180 - 31=149^{\circ}\). \(\angle 4\) and \(\angle 3\) are supplementary (linear pair), so \(m\angle 4=180 - 31 = 149^{\circ}\). And \(\angle 2\) and \(\angle 4\) are vertical angles, so they are equal. That makes sense.

Answer:

\(m\angle 2=\boxed{149}\) degrees, \(m\angle 4=\boxed{149}\) degrees