Question

for the figure below, give the following.
(a) one pair of vertical angles
(b) one pair of angles that form a linear pair
(c) one pair of angles that are supplementary.

Explanation:

Step1: Recall vertical - angle definition

Vertical angles are opposite angles formed by the intersection of two lines. For the intersection of lines \(l\) and \(n\), \(\angle2\) and \(\angle3\) are vertical angles (also \(\angle1\) and \(\angle4\)). For the intersection of lines \(m\) and \(n\), \(\angle6\) and \(\angle7\) are vertical angles (also \(\angle5\) and \(\angle8\)). Let's choose \(\angle2\) and \(\angle3\).

Step2: Recall linear - pair definition

A linear pair of angles are adjacent angles that form a straight - line. \(\angle1\) and \(\angle2\) form a linear pair (also \(\angle2\) and \(\angle4\), \(\angle4\) and \(\angle3\), \(\angle3\) and \(\angle1\); and for the other intersection, \(\angle5\) and \(\angle6\), \(\angle6\) and \(\angle8\), \(\angle8\) and \(\angle7\), \(\angle7\) and \(\angle5\)). Let's choose \(\angle1\) and \(\angle2\).

Step3: Recall supplementary - angle definition

Supplementary angles are two angles whose sum is \(180^{\circ}\). A linear pair of angles are always supplementary. Since \(\angle1\) and \(\angle2\) form a linear pair, they are supplementary. We can also choose non - linear - pair supplementary angles if the lines are parallel and we consider same - side interior angles etc. But a linear pair satisfies the supplementary condition. Let's choose \(\angle1\) and \(\angle2\) again.

Answer:

(a) \(\angle2\) and \(\angle3\)
(b) \(\angle1\) and \(\angle2\)
(c) \(\angle1\) and \(\angle2\)