Question

1. is the statement sometimes, always, or never true?

image of two intersecting lines with angles labeled 1, 4, 3, 2 (in order around the intersection)
if ( mangle 1 = 40^circ ), then ( mangle 2 = 140^circ ).
a. sometimes
b. always
c. never

Explanation:

Step1: Identify angle relationships

Vertical angles are equal, and linear pairs are supplementary (sum to \(180^\circ\)). \(\angle1\) and \(\angle2\): \(\angle1\) and \(\angle3\) are supplementary, \(\angle3\) and \(\angle2\) are vertical (equal). Wait, no—\(\angle1\) and \(\angle2\): actually, \(\angle1\) and \(\angle2\) are... Wait, looking at the diagram, \(\angle1\) and \(\angle3\) form a linear pair, \(\angle3\) and \(\angle2\) are vertical angles (so \(\angle3=\angle2\))? No, wait, vertical angles: \(\angle1\) and \(\angle2\)? Wait, no, the two lines intersect, so \(\angle1\) and \(\angle2\) are vertical angles? Wait, no, let's label: when two lines intersect, opposite angles are vertical. So \(\angle1\) and \(\angle2\) are vertical angles? Wait, no, in the diagram, \(\angle1\) and \(\angle2\): wait, maybe I mislabel. Wait, the lines are intersecting, so \(\angle1\) and \(\angle3\) are adjacent (linear pair), \(\angle3\) and \(\angle2\) are vertical? No, \(\angle1\) and \(\angle2\) are vertical angles? Wait, no, let's see: \(\angle1\) and \(\angle2\) are opposite? Wait, maybe the diagram has \(\angle1\) and \(\angle2\) as vertical angles? Wait, no, if \(m\angle1 = 40^\circ\), and \(\angle2\) is... Wait, no, maybe \(\angle1\) and \(\angle2\) are not vertical. Wait, maybe the problem is: \(\angle1\) and \(\angle2\): wait, the statement says if \(m\angle1 = 40^\circ\), then \(m\angle2 = 140^\circ\). But vertical angles are equal, so if \(\angle1\) and \(\angle2\) are vertical, then \(m\angle2 = 40^\circ\), not \(140^\circ\). Wait, but maybe I misread the diagram. Wait, maybe \(\angle1\) and \(\angle2\) are supplementary? No, vertical angles are equal. Wait, no—wait, maybe the diagram is two intersecting lines, so \(\angle1\) and \(\angle4\) are supplementary, \(\angle4\) and \(\angle2\) are supplementary? No, let's recall: when two lines intersect, vertical angles are equal, and linear pairs (adjacent angles on a straight line) are supplementary (sum to \(180^\circ\)). So \(\angle1\) and \(\angle3\) are a linear pair (\(m\angle1 + m\angle3 = 180^\circ\)), \(\angle3\) and \(\angle2\) are vertical angles (\(m\angle3 = m\angle2\)). Therefore, \(m\angle1 + m\angle2 = 180^\circ\)? Wait, no: \(\angle1\) and \(\angle3\) are linear pair, so \(m\angle1 + m\angle3 = 180^\circ\). \(\angle3\) and \(\angle2\) are vertical, so \(m\angle3 = m\angle2\). Therefore, \(m\angle1 + m\angle2 = 180^\circ\)? Wait, that would mean \(m\angle2 = 180^\circ - m\angle1\). If \(m\angle1 = 40^\circ\), then \(m\angle2 = 140^\circ\). Wait, but that would be if \(\angle1\) and \(\angle2\) are supplementary. But are they? Wait, maybe the diagram shows \(\angle1\) and \(\angle2\) as supplementary. Wait, no, vertical angles are equal. Wait, I think I made a mistake. Let's re-express: two intersecting lines form vertical angles (equal) and linear pairs (supplementary). So \(\angle1\) and \(\angle2\): if \(\angle1\) and \(\angle2\) are vertical angles, then they are equal. But the statement says \(m\angle2 = 140^\circ\) when \(m\angle1 = 40^\circ\), which are supplementary. So maybe \(\angle1\) and \(\angle2\) are not vertical, but... Wait, maybe the diagram is such that \(\angle1\) and \(\angle2\) are supplementary. Wait, no, the problem's diagram: let's assume that \(\angle1\) and \(\angle2\) are vertical angles? No, that can't be. Wait, maybe the question is misphrased, or I missee the diagram. Wait, the options are sometimes, always, never. Wait, if \(\angle1\) and \(\angle2\) are vertical angles, then \(m\angle2 = 40^\circ\), so the statement would be never true. But that contradicts. Wait, no—wait, maybe \(\angle1\) and \(\angle2\) are adjacent and form a linear pair? No, vertical angles are opposite. Wait, maybe the diagram is two lines intersecting, so \(\angle1\) and \(\angle2\) are vertical angles, so they should be equal. Therefore, if \(m\angle1 = 40^\circ\), \(m\angle2\) should be \(40^\circ\), not \(140^\circ\). Therefore, the statement "if \(m\angle1 = 40^\circ\), then \(m\angle2 = 140^\circ\)" is never true? But that seems wrong. Wait, no—wait, maybe I mixed up the angles. Wait, maybe \(\angle1\) and \(\angle4\) are supplementary, \(\angle4\) and \(\angle2\) are vertical? No, \(\angle4\) and \(\angle2\) would be vertical? Wait, no, when two lines intersect, the vertical angles are \(\angle1\) & \(\angle2\), \(\angle3\) & \(\angle4\)? No, that's not right. Wait, standard intersection: angle 1 and angle 3 are vertical? No, no—let's label the intersection: top angle \(\angle4\), left \(\angle1\), bottom \(\angle3\), right \(\angle2\). So \(\angle1\) and \(\angle3\) are vertical? No, \(\angle1\) and \(\angle2\) are vertical? Wait, no, vertical angles are opposite each other. So \(\angle1\) (left) and \(\angle2\) (right) are vertical? No, \(\angle1\) (left) and \(\angle3\) (bottom) are adjacent, \(\angle3\) (bottom) and \(\angle2\) (right) are adjacent, \(\angle2\) (right) and \(\angle4\) (top) are adjacent, \(\angle4\) (top) and \(\angle1\) (left) are adjacent. So vertical angles: \(\angle1\) and \(\angle2\) are not vertical. Wait, \(\angle1\) and \(\angle2\) are actually vertical? No, that's not possible. Wait, maybe the diagram is two lines intersecting, so \(\angle1\) and \(\angle2\) are vertical angles, so they are equal. Therefore, if \(m\angle1 = 40^\circ\), \(m\angle2 = 40^\circ\), so the statement \(m\angle2 = 140^\circ\) is never true. But that would mean option C. But that seems conflicting. Wait, no—wait, maybe I made a mistake. Wait, let's check the linear pair. \(\angle1\) and \(\angle4\) are a linear pair, so \(m\angle1 + m\angle4 = 180^\circ\). \(\angle4\) and \(\angle2\) are vertical angles, so \(m\angle4 = m\angle2\). Therefore, \(m\angle1 + m\angle2 = 180^\circ\). Oh! Wait, that's the key. So \(\angle4\) and \(\angle2\) are vertical angles (equal), and \(\angle1\) and \(\angle4\) are supplementary (sum to \(180^\circ\)). Therefore, \(m\angle1 + m\angle2 = 180^\circ\) (because \(m\angle4 = m\angle2\)). So if \(m\angle1 = 40^\circ\), then \(m\angle2 = 180^\circ - 40^\circ = 140^\circ\). Therefore, the statement is always true. So the answer is B.

Step2: Verify the relationship

Since \(\angle1\) and \(\angle4\) are supplementary (\(m\angle1 + m\angle4 = 180^\circ\)) and \(\angle4\) and \(\angle2\) are vertical angles (\(m\angle4 = m\angle2\)), substituting gives \(m\angle1 + m\angle2 = 180^\circ\). For \(m\angle1 = 40^\circ\), \(m\angle2 = 180^\circ - 40^\circ = 140^\circ\), so the statement holds always.

Answer:

B. always