Question

determine whether the given angles are a linear pair, vertical angles, or neither. diagram: lines vq r (horizontal), pq (left - up), u, t, s (right - down), angles 1 (between pq and vq), 2 (between vq and u), 3, 4 (between u, t, s), 5 (between s and qr), 6 (between pq and qr). angle relationships: ∠1 and ∠6, ∠2 and ∠5, ∠1 and ∠5. options: linear pair, vertical angles, neither (drag & drop the answer)

Explanation:

For $\boldsymbol{\angle 1}$ and $\boldsymbol{\angle 6}$:

Step1: Recall linear pair definition

A linear pair of angles are adjacent and supplementary (sum to $180^\circ$), forming a straight line.
$\angle 1$ and $\angle 6$ share a common side and vertex, and their non - common sides form a straight line (since $VP$ and $QR$ are on a straight line? Wait, actually, $\angle 1$ and $\angle 6$ are adjacent and their sum is $180^\circ$ as they form a linear pair. Wait, no, wait. Wait, $\angle 1$ and $\angle 6$: looking at the diagram, $VQ R$ is a straight line? Wait, $VQ$ and $QR$ are a straight line (since $V$, $Q$, $R$ are colinear). $\angle 1$ and $\angle 6$ are adjacent angles that form a straight line, so they are a linear pair. Wait, no, wait, maybe I made a mistake. Wait, $\angle 1$ and $\angle 6$: if $VP$ and $VQ$? Wait, no, the diagram has $V$, $Q$, $R$ on a straight line, and $P$, $Q$, $...$ Wait, maybe $\angle 1$ and $\angle 6$: their sum is $180^\circ$ and they are adjacent, so linear pair.

Step2: Classify $\angle 1$ and $\angle 6$

Since $\angle 1$ and $\angle 6$ are adjacent, form a straight line (sum to $180^\circ$), they are a linear pair.

For $\boldsymbol{\angle 2}$ and $\boldsymbol{\angle 5}$:

Step1: Recall neither definition

Check if they are linear pair (adjacent, sum to $180^\circ$) or vertical angles (opposite, equal). $\angle 2$ and $\angle 5$: they are not adjacent, and not opposite angles formed by two intersecting lines. So they are neither.

Step2: Classify $\angle 2$ and $\angle 5$

Since they don't meet the criteria for linear pair or vertical angles, they are neither.

For $\boldsymbol{\angle 1}$ and $\boldsymbol{\angle 5}$:

Step1: Recall vertical angles definition

Vertical angles are opposite angles formed by two intersecting lines, and they are equal. $\angle 1$ and $\angle 5$: when lines $VP$ and $QR$ intersect at $Q$, $\angle 1$ and $\angle 5$ are opposite angles. So they are vertical angles.

Step2: Classify $\angle 1$ and $\angle 5$

Since they are opposite angles formed by intersecting lines, they are vertical angles.

Answer:

• $\angle 1$ and $\angle 6$: linear pair
• $\angle 2$ and $\angle 5$: neither
• $\angle 1$ and $\angle 5$: vertical angles