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

Explanation:

Part (a)

Step1: Recall vertical angles definition

Vertical angles are opposite angles formed by two intersecting lines, and they are congruent. Looking at the figure, lines \( l \) and \( n \) intersect, forming angles \( \angle 1, \angle 2, \angle 3, \angle 4 \). Also, lines \( m \) and \( n \) intersect, forming angles \( \angle 5, \angle 6, \angle 7, \angle 8 \). For example, \( \angle 1 \) and \( \angle 3 \) are vertical angles (formed by intersection of \( l \) and \( n \)). Another example: \( \angle 2 \) and \( \angle 4 \), or \( \angle 5 \) and \( \angle 7 \), or \( \angle 6 \) and \( \angle 8 \). Let's take \( \angle 1 \) and \( \angle 3 \).

Step2: Confirm vertical angles property

Since \( l \) and \( n \) intersect, \( \angle 1 \) and \( \angle 3 \) are opposite angles, so they are vertical angles.

Part (b)

Step1: Recall linear pair definition

A linear pair of angles are adjacent angles that form a straight line (sum to \( 180^\circ \)). Looking at the figure, for example, \( \angle 2 \) and \( \angle 3 \) are adjacent and form a straight line (since they are on line \( n \) with line \( l \) intersecting it). Another example: \( \angle 1 \) and \( \angle 2 \), \( \angle 3 \) and \( \angle 4 \), \( \angle 5 \) and \( \angle 6 \), \( \angle 7 \) and \( \angle 8 \), etc. Let's take \( \angle 2 \) and \( \angle 3 \).

Step2: Confirm linear pair property

\( \angle 2 \) and \( \angle 3 \) are adjacent, share a common side, and their non - common sides form a straight line (line \( n \)), so they form a linear pair.

Part (c)

Step1: Recall congruent angles (vertical angles are congruent)

From part (a), vertical angles are congruent. So we can use the vertical angles we found in part (a). For example, \( \angle 1 \) and \( \angle 3 \) are congruent (vertical angles). Also, if we consider the vertical angles from the intersection of \( m \) and \( n \), like \( \angle 5 \) and \( \angle 7 \), or \( \angle 6 \) and \( \angle 8 \). Let's take \( \angle 1 \) and \( \angle 3 \) (same as part (a) since vertical angles are congruent).

Step2: Confirm congruence

Since \( \angle 1 \) and \( \angle 3 \) are vertical angles, by the vertical angles theorem, they are congruent.

Answer:

s:
(a) Vertical angles: \( \angle 1 \) and \( \angle 3 \) (or other valid vertical angle pairs like \( \angle 2 \) and \( \angle 4 \), \( \angle 5 \) and \( \angle 7 \), \( \angle 6 \) and \( \angle 8 \))
(b) Linear pair: \( \angle 2 \) and \( \angle 3 \) (or other valid linear pair like \( \angle 1 \) and \( \angle 2 \), \( \angle 3 \) and \( \angle 4 \), \( \angle 5 \) and \( \angle 6 \), \( \angle 7 \) and \( \angle 8 \))
(c) Congruent angles: \( \angle 1 \) and \( \angle 3 \) (or other valid congruent angle pairs like \( \angle 2 \) and \( \angle 4 \), \( \angle 5 \) and \( \angle 7 \), \( \angle 6 \) and \( \angle 8 \))