Question

there is a diagram with lines intersecting at point u, forming angles labeled 6,7,8,9,10. there is a table with angle pairs (∠6 and ∠9, ∠8 and ∠9, ∠7 and ∠8, ∠6 and ∠8) and columns (complementary, supplementary, vertical, adjacent, none) with checkboxes.

Explanation:

To solve this, we analyze each pair of angles using the definitions of complementary, supplementary, vertical, and adjacent angles:

1. $\boldsymbol{\angle 6}$ and $\boldsymbol{\angle 9}$

• Complementary: Sum to $90^\circ$? $\angle 10$ is a right angle ($90^\circ$), but $\angle 6$ and $\angle 9$: $\angle 6 + \angle 9$? From the diagram, $\angle 6$ and $\angle 9$ – no, $\angle 9 + \angle 10 = 90^\circ$? Wait, $\angle 9$ and $\angle 10$ are complementary? Wait, no, let's re - examine. $\angle 10$ is a right angle. $\angle 6$ and $\angle 9$: Are they vertical? No. Adjacent? No. Supplementary? Sum to $180^\circ$? No. Complementary? No. Wait, maybe I made a mistake. Wait, $\angle 6$ and $\angle 8$ are vertical? Wait, let's go step by step for each pair.

2. $\boldsymbol{\angle 8}$ and $\boldsymbol{\angle 9}$

• Adjacent: They share a common side (the ray from $U$) and a common vertex ($U$), and their non - common sides are adjacent. Also, $\angle 8+\angle 9$: Do they sum to $180^\circ$? No. Complementary? No. Vertical? No. So adjacent.

3. $\boldsymbol{\angle 7}$ and $\boldsymbol{\angle 8}$

• Supplementary: They form a linear pair (they are adjacent and their non - common sides form a straight line), so their sum is $180^\circ$. So they are supplementary.

4. $\boldsymbol{\angle 6}$ and $\boldsymbol{\angle 8}$

• Vertical Angles: They are opposite each other when two lines intersect. So $\angle 6$ and $\angle 8$ are vertical angles.

Let's fill the table:

Pair of Angles	Complementary	Supplementary	Vertical	Adjacent	None
$\angle 8$ and $\angle 9$	$\square$	$\square$	$\square$	$\checkmark$	$\square$
$\angle 7$ and $\angle 8$	$\square$	$\checkmark$	$\square$	$\checkmark$ (since they are adjacent and supplementary)	$\square$
$\angle 6$ and $\angle 8$	$\square$	$\square$	$\checkmark$	$\square$	$\square$

If we assume the question is to mark the correct relationship for each pair:

• For $\angle 6$ and $\angle 9$: None (mark the "None" column)
• For $\angle 8$ and $\angle 9$: Adjacent (mark the "Adjacent" column)
• For $\angle 7$ and $\angle 8$: Supplementary (and adjacent, but mainly supplementary as a linear pair) (mark the "Supplementary" column)
• For $\angle 6$ and $\angle 8$: Vertical (mark the "Vertical" column)

If we take a single pair, for example, if we consider $\angle 6$ and $\angle 8$:

Step 1: Recall the definition of vertical angles

Vertical angles are the opposite angles formed when two lines intersect. They are equal in measure.

Step 2: Analyze $\angle 6$ and $\angle 8$

In the diagram, the two lines (the one with $F - U - N$ and the one with $C - U - V$) intersect at $U$. $\angle 6$ and $\angle 8$ are opposite each other, so they are vertical angles.

Step 1: Recall the definition of supplementary angles

Supplementary angles are two angles whose sum is $180^\circ$.

Step 2: Analyze $\angle 7$ and $\angle 8$

$\angle 7$ and $\angle 8$ form a linear pair (they are adjacent and their non - common sides form a straight line). So $\angle 7+\angle 8 = 180^\circ$, which means they are supplementary.

Step 1: Recall the definition of adjacent angles

Adjacent angles are two angles that share a common side and a common vertex, and their non - common sides are adjacent.

Step 2: Analyze $\angle 8$ and $\angle 9$

$\angle 8$ and $\angle 9$ share the vertex $U$ and a common side (the ray from $U$), and their non - common sides are adjacent. So they are adjacent angles.

Answer:

For $\angle 6$ and $\angle 8$: Vertical Angles.

(If we consider $\angle 7$ and $\angle 8$: