Question

check for understanding: parallel lines and the transversal
not graded
directions
answer the question below to check your understanding and move forward in the course! this quiz is not graded.
true or false? there are four linear pairs and four vertical angle pairs in this figure.
image of a figure with angles labeled 1,2,3,4 and a transversal

Explanation:

Step1: Recall Linear Pair Definition

A linear pair is two adjacent angles that form a straight line (sum to \(180^\circ\)). Let's identify linear pairs: \(\angle1\) & \(\angle2\), \(\angle2\) & \(\angle3\), \(\angle3\) & \(\angle4\), \(\angle4\) & \(\angle1\)? Wait, no—wait the figure: horizontal line and a transversal. Wait, the horizontal line has angles: let's label the horizontal line as \(l\), transversal as \(m\). Angles: \(\angle1\) (top right), \(\angle2\) (top left), \(\angle3\) (bottom left), \(\angle4\) (bottom right), and the straight line (so \(\angle2\) & the angle adjacent to it on the straight line? Wait, maybe the figure has a straight horizontal line and a transversal, creating four angles at the intersection, plus another straight segment? Wait, no—the figure shows a horizontal line (with a straight extension, so a straight line) and a transversal intersecting it, creating four angles: \(\angle1\), \(\angle2\), \(\angle3\), \(\angle4\), and the horizontal line is straight, so \(\angle2\) and the angle to its left (let's say \(\angle5\) and \(\angle6\)?) Wait, no, the user's figure: "1, 2, 3, 4" and a straight horizontal line. Wait, maybe the horizontal line is a straight line, so two linear pairs on the transversal intersection: \(\angle1\) & \(\angle2\), \(\angle2\) & \(\angle3\), \(\angle3\) & \(\angle4\), \(\angle4\) & \(\angle1\)? No, linear pairs are adjacent and supplementary. At the intersection of two lines (transversal and horizontal), there are four angles: \(\angle1\) (top right), \(\angle2\) (top left), \(\angle3\) (bottom left), \(\angle4\) (bottom right). Then, \(\angle1\) and \(\angle2\) are adjacent (linear pair), \(\angle2\) and \(\angle3\) (linear pair), \(\angle3\) and \(\angle4\) (linear pair), \(\angle4\) and \(\angle1\) (linear pair)? Wait, no—at a straight line (horizontal), the transversal creates two linear pairs on top: \(\angle1 + \angle2 = 180^\circ\), and two on bottom: \(\angle3 + \angle4 = 180^\circ\)? No, that's not right. Wait, linear pair: two angles that are adjacent (share a common side) and their non - common sides form a straight line. So at the intersection of the transversal and the horizontal line, the four angles: \(\angle1\) (top right), \(\angle2\) (top left), \(\angle3\) (bottom left), \(\angle4\) (bottom right). Then, \(\angle1\) and \(\angle2\) are adjacent (linear pair, sum \(180^\circ\)), \(\angle2\) and \(\angle3\) (linear pair, sum \(180^\circ\)), \(\angle3\) and \(\angle4\) (linear pair, sum \(180^\circ\)), \(\angle4\) and \(\angle1\) (linear pair, sum \(180^\circ\))? Wait, no, that's four linear pairs. Now vertical angles: \(\angle1\) & \(\angle3\), \(\angle2\) & \(\angle4\) – that's two vertical angle pairs. Wait, but the question says four linear pairs and four vertical angle pairs. Wait, maybe the horizontal line is a straight line (so a straight angle), and the transversal intersects it, creating two intersections? No, the figure is likely a transversal intersecting a straight line (so one intersection), creating four angles, but the straight line has a linear pair on each side? Wait, no—maybe the figure has a straight horizontal line (so a straight angle, \(180^\circ\)) and a transversal intersecting it, creating four angles at the intersection, and the straight line has two linear pairs (e.g., \(\angle2\) and the angle to its left, \(\angle1\) and the angle to its right), but that's confusing. Wait, the key: vertical angles are opposite angles when two lines intersect, so only two pairs (\(\angle1\) & \(\angle3\), \(\angle2\) & \(\angle4\)). Linear pairs: at the intersection, four linear pairs? No, at two - line intersection, there are four linear pairs? Wait, no: when two lines intersect, there are four angles. Each angle forms a linear pair with two adjacent angles? No, each angle has one linear pair. Wait, \(\angle1\) and \(\angle2\) (linear pair), \(\angle2\) and \(\angle3\) (linear pair), \(\angle3\) and \(\angle4\) (linear pair), \(\angle4\) and \(\angle1\) (linear pair) – that's four linear pairs. Vertical angles: \(\angle1\) & \(\angle3\), \(\angle2\) & \(\angle4\) – two pairs. But the question says four vertical angle pairs, which is wrong. Wait, maybe the figure has two intersections? Like a transversal intersecting two parallel lines? No, the title is "Parallel Lines and the Transversal", but the question is about linear pairs and vertical angle pairs. Wait, no—if it's a transversal intersecting a straight line (one line), then two vertical angle pairs, four linear pairs? Wait, no, vertical angles are only two pairs when two lines intersect. So the statement "four vertical angle pairs" is false. Wait, but let's re - check: linear pairs: at the intersection of two lines, each angle has one linear pair, so four angles, four linear pairs (each angle + adjacent angle = \(180^\circ\)). Vertical angles: two pairs (opposite angles). So the statement says "four linear pairs and four vertical angle pairs" – vertical angle pairs are two, not four. So the answer is False.

Step2: Recall Vertical Angle Definition

Vertical angles are opposite angles formed by two intersecting lines, so there are two pairs (\(\angle1\) & \(\angle3\), \(\angle2\) & \(\angle4\)) when two lines intersect. The question claims four vertical angle pairs, which is incorrect. Linear pairs: at the intersection, there are four linear pairs (as each angle is adjacent to one angle to form a linear pair, and there are four angles, so four linear pairs). But the vertical angle part is wrong, so the statement is False.

Answer:

False