Question

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

Explanation:

(a) Linear Pair

Step1: Recall linear pair definition

A linear pair of angles are adjacent and form a straight line (sum to \(180^\circ\)).

Step2: Identify adjacent angles on a line

\(\angle 1\) and \(\angle 2\) are adjacent, share a common side, and their non - common sides form a straight line. So \(\angle 1\) and \(\angle 2\) form a linear pair. (Other valid pairs: \(\angle 2\) & \(\angle 4\), \(\angle 4\) & \(\angle 3\), \(\angle 3\) & \(\angle 1\), \(\angle 5\) & \(\angle 6\), etc.)

Step1: Recall vertical angles definition

Vertical angles are opposite angles formed by intersecting lines, and they are equal.

Step2: Identify opposite angles

When two lines intersect, \(\angle 1\) and \(\angle 4\) are opposite angles formed by the intersection of lines \(l\) and \(n\). (Other valid pairs: \(\angle 2\) & \(\angle 3\), \(\angle 5\) & \(\angle 8\), \(\angle 6\) & \(\angle 7\))

Step1: Recall supplementary angles definition

Supplementary angles sum to \(180^\circ\). Linear pairs are supplementary, but also non - adjacent angles can be supplementary if their sum is \(180^\circ\).

Step2: Find supplementary angles

\(\angle 1\) and \(\angle 5\) (if lines are parallel, but even without parallelism, \(\angle 1\) and \(\angle 2\) are supplementary as a linear pair, or \(\angle 1\) and \(\angle 4\) are not supplementary, but \(\angle 1\) and \(\angle 3\) are supplementary (linear pair), or \(\angle 5\) and \(\angle 7\) (linear pair). Another example: \(\angle 1\) and \(\angle 5\) (if we consider transversal, but a simple linear pair like \(\angle 1\) and \(\angle 2\) is supplementary. Let's take \(\angle 1\) and \(\angle 2\) (same as linear pair, since linear pairs are supplementary).

Answer:

\(\angle 1\) and \(\angle 2\) (or other valid linear pair like \(\angle 2\) and \(\angle 4\) etc.)

(b) Vertical Angles