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

Explanation:

Part (a)

Step1: Recall linear pair definition

A linear pair of angles are adjacent and supplementary (form a straight line, sum to \(180^\circ\)). Looking at the figure, \(\angle 1\) and \(\angle 2\) are adjacent and form a straight line (along line \(l\)).

Step2: Identify the pair

So one linear pair is \(\angle 1\) and \(\angle 2\) (other possible pairs: \(\angle 2\) & \(\angle 3\), \(\angle 3\) & \(\angle 4\), \(\angle 4\) & \(\angle 1\), \(\angle 5\) & \(\angle 6\), etc.)

Part (b)

Step1: Recall vertical angles definition

Vertical angles are opposite angles formed by two intersecting lines, and they are congruent. For lines \(l\) and \(n\) intersecting, \(\angle 1\) and \(\angle 3\) are vertical angles (opposite each other).

Step2: Identify the pair

So one pair of vertical angles is \(\angle 1\) and \(\angle 3\) (other possible pairs: \(\angle 2\) & \(\angle 4\), \(\angle 5\) & \(\angle 7\), \(\angle 6\) & \(\angle 8\))

Part (c)

Step1: Recall congruent angles (vertical angles are congruent)

From vertical angles, \(\angle 1\) and \(\angle 3\) are congruent. Also, if lines \(l\) and \(m\) are parallel (implied by the figure's structure, though not stated, vertical angles or corresponding angles can be congruent), but using vertical angles is straightforward.

Step2: Identify the pair

So one pair of congruent angles is \(\angle 1\) and \(\angle 3\) (or \(\angle 2\) & \(\angle 4\), \(\angle 5\) & \(\angle 7\), \(\angle 6\) & \(\angle 8\))

Answer:

s:
(a) \(\angle 1\) and \(\angle 2\) (or other valid linear pair like \(\angle 2\) & \(\angle 3\), etc.)
(b) \(\angle 1\) and \(\angle 3\) (or other valid vertical pair like \(\angle 2\) & \(\angle 4\), etc.)
(c) \(\angle 1\) and \(\angle 3\) (or other valid congruent pair like vertical angles or corresponding angles if \(l\parallel m\))

(Note: For (a), any adjacent angles forming a straight line; for (b), opposite angles from intersecting lines; for (c), vertical angles or corresponding angles if parallel. The above are example answers.)