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

(a) linear pair: ∠ and ∠
(b) vertical angles: ∠ and ∠
(c) supplementary angles: ∠ and ∠

Explanation:

Part (a)

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 and form a straight line (they share a common side and their non - common sides are opposite rays).

Part (b)

Step1: Recall vertical angles definition

Vertical angles are opposite angles formed by the intersection of two lines.

Step2: Identify vertical angles

When two lines intersect, like the intersection of line \(l\) and the horizontal line, \(\angle 1\) and \(\angle 3\) are vertical angles (or \(\angle 2\) and \(\angle 4\), or \(\angle 5\) and \(\angle 7\), or \(\angle 6\) and \(\angle 8\)). Let's take \(\angle 1\) and \(\angle 3\) as an example.

Part (c)

Step1: Recall supplementary angles definition

Supplementary angles are two angles whose sum is \(180^\circ\). Linear pairs are supplementary, and also non - adjacent angles that sum to \(180^\circ\) are supplementary.

Step2: Identify supplementary angles

Using the linear pair idea, \(\angle 1\) and \(\angle 2\) are supplementary (as they are a linear pair). Also, for example, \(\angle 3\) and \(\angle 4\) are supplementary, or \(\angle 5\) and \(\angle 6\) are supplementary. Let's take \(\angle 1\) and \(\angle 2\) as an example.

Answer:

s:
(a) Linear pair: \(\angle 1\) and \(\angle 2\) (other possible pairs: \(\angle 2\) and \(\angle 3\), \(\angle 3\) and \(\angle 4\), \(\angle 4\) and \(\angle 1\), \(\angle 5\) and \(\angle 6\), \(\angle 6\) and \(\angle 7\), \(\angle 7\) and \(\angle 8\), \(\angle 8\) and \(\angle 5\))

(b) Vertical angles: \(\angle 1\) and \(\angle 3\) (other possible pairs: \(\angle 2\) and \(\angle 4\), \(\angle 5\) and \(\angle 7\), \(\angle 6\) and \(\angle 8\))

(c) Supplementary angles: \(\angle 1\) and \(\angle 2\) (other possible pairs: \(\angle 2\) and \(\angle 3\), \(\angle 3\) and \(\angle 4\), \(\angle 4\) and \(\angle 1\), \(\angle 5\) and \(\angle 6\), \(\angle 6\) and \(\angle 7\), \(\angle 7\) and \(\angle 8\), \(\angle 8\) and \(\angle 5\), also non - linear pair supplementary angles like \(\angle 3\) and \(\angle 6\) if the lines are such that their sum is \(180^\circ\))