Question

give one pair of supplementary angles and one pair of vertical angles shown in the figure below.

Explanation:

Step1: Recall Supplementary Angles

Supplementary angles are two angles whose sum is \(180^\circ\). In the figure, adjacent angles on a straight line are supplementary. For example, \(\angle1\) and \(\angle2\) are adjacent and form a linear pair, so \(\angle1 + \angle2 = 180^\circ\), so \(\angle1\) and \(\angle2\) are supplementary.

Step2: Recall Vertical Angles

Vertical angles are opposite angles formed by the intersection of two lines. They are equal. For example, \(\angle1\) and \(\angle5\) are vertical angles (wait, no, looking at the labels: when two lines intersect, vertical angles are opposite. Let's see the labels: the two intersecting lines, so \(\angle1\) and \(\angle5\)? Wait, no, maybe \(\angle1\) and \(\angle3\)? Wait, no, the first intersection: the horizontal line and the vertical line (labeled l) and the slant line (labeled m). Wait, maybe \(\angle1\) and \(\angle2\) are supplementary (linear pair), and vertical angles: \(\angle1\) and \(\angle5\)? No, maybe \(\angle2\) and \(\angle6\)? Wait, let's correct: when two lines intersect, vertical angles are opposite. So for the horizontal and vertical line (l), \(\angle1\) and \(\angle5\) are vertical? No, maybe the slant line (m) and horizontal line: \(\angle1\) and \(\angle3\)? Wait, maybe better: supplementary angles: any two adjacent angles on a straight line. So \(\angle1\) and \(\angle2\) (sum to \(180^\circ\)). Vertical angles: \(\angle1\) and \(\angle5\) (no, maybe \(\angle2\) and \(\angle6\)? Wait, the labels are 1,2,3,4,5,6,7,8. Let's assume the horizontal line is intersected by line l (vertical) and line m (slant). So line l and horizontal: angles 1,2,5,6. Line m and horizontal: angles 3,4,7,8. So supplementary angles: \(\angle1\) and \(\angle2\) (linear pair, sum \(180^\circ\)). Vertical angles: \(\angle1\) and \(\angle5\) (no, \(\angle2\) and \(\angle6\) are vertical? Wait, no, vertical angles are opposite when two lines cross. So if line l (vertical) crosses horizontal, then \(\angle1\) and \(\angle5\) are vertical? No, \(\angle1\) and \(\angle5\) are same side? Wait, maybe the labels are: at the intersection of line l and horizontal: \(\angle1\) (top right), \(\angle2\) (top left), \(\angle5\) (bottom right), \(\angle6\) (bottom left). At intersection of line m and horizontal: \(\angle3\) (top right between line l and m), \(\angle4\) (top left of line m), \(\angle7\) (bottom right of line m), \(\angle8\) (bottom left of line m). So supplementary angles: \(\angle1\) and \(\angle2\) (sum \(180^\circ\)). Vertical angles: \(\angle1\) and \(\angle5\) (no, \(\angle2\) and \(\angle6\) are vertical (opposite, equal). Wait, maybe \(\angle1\) and \(\angle2\) are supplementary, and vertical angles: \(\angle2\) and \(\angle6\).

Answer:

(a) Supplementary angles: \(\angle1\) and \(\angle2\) (or any linear pair, e.g., \(\angle3\) and \(\angle4\), etc.)
(b) Vertical angles: \(\angle2\) and \(\angle6\) (or \(\angle1\) and \(\angle5\), \(\angle3\) and \(\angle7\), \(\angle4\) and \(\angle8\))

(Note: The specific labels may vary, but the concept is linear pair for supplementary, opposite for vertical.)