Question

18 name each pair of vertical angles image of two parallel lines cut by a transversal, with angles labeled 1,2,3,4 (top line), 5,6,7,8 (bottom line) a 1 & 3 b 1 & 2 c 1 & 4 d 1 & 5 e 6 & 8 f 5 & 7 g 7 & 8 h 2 & 8 i 3 & 4 j 2 & 4

Explanation:

Step1: Recall Vertical Angles Definition

Vertical angles are opposite angles formed by two intersecting lines, and they are equal.

Step2: Analyze Each Option

- Option A: ∠1 and ∠3 are adjacent, not vertical.
- Option B: ∠1 and ∠2 are adjacent (form a linear pair), not vertical.
- Option C: ∠1 and ∠4 are adjacent, not vertical.
- Option D: ∠1 and ∠5 are not formed by intersecting lines (different transversal interaction), not vertical.
- Option E: ∠6 and ∠8: When lines intersect, ∠6 and ∠8 are opposite (vertical angles).
- Option F: ∠5 and ∠7: Opposite angles from intersecting lines, vertical angles.
- Option G: ∠7 and ∠8 are adjacent, not vertical.
- Option H: ∠2 and ∠8: Not vertical (different positions).
- Option I: ∠3 and ∠4 are adjacent, not vertical.
- Option J: ∠2 and ∠4: Opposite angles from intersecting lines, vertical angles.

Wait, let's re - check the diagram. The two horizontal lines and one transversal. Let's list all vertical angle pairs:
- From the top intersection (of the transversal and top horizontal line): ∠1 and ∠3? No, wait, ∠1 and ∠3 are adjacent? Wait, no, at the top intersection, the angles are ∠1 (top - left), ∠2 (top - right), ∠4 (bottom - left), ∠3 (bottom - right). So vertical angles here: ∠1 and ∠3? No, ∠1 and ∠3 are adjacent? Wait, no, vertical angles are opposite. So ∠1 and ∠3: no, ∠1 and ∠4? No, ∠2 and ∠4? Wait, ∠1 (top - left) and ∠3 (bottom - right) are vertical? Wait, no, when two lines intersect, the vertical angles are opposite. So at the top intersection: ∠1 (top, left) and ∠3 (bottom, right) are vertical? Or ∠2 (top, right) and ∠4 (bottom, left) are vertical. Then at the bottom intersection (transversal and bottom horizontal line): ∠5 (top - left), ∠6 (top - right), ∠8 (bottom - left), ∠7 (bottom - right). So vertical angles here: ∠5 and ∠7, ∠6 and ∠8. Also, ∠2 (top - right of top intersection) and ∠4 (bottom - left of top intersection) are vertical? Wait, maybe I made a mistake earlier.

Let's correctly identify vertical angles:
- At the top intersection (transversal and upper horizontal line):
- ∠1 (top - left) and ∠3 (bottom - right) are vertical? No, wait, ∠1 (top - left) and ∠3 (bottom - right) are actually vertical? Wait, no, when two lines intersect, the vertical angles are the pairs that are opposite. So if we have two intersecting lines, forming four angles: let's call them A (top - left), B (top - right), C (bottom - right), D (bottom - left). Then vertical angles are A and C, B and D. So in the top intersection: ∠1 (A), ∠2 (B), ∠4 (D), ∠3 (C). So vertical angles: ∠1 and ∠3, ∠2 and ∠4.

At the bottom intersection (transversal and lower horizontal line): angles ∠5 (A), ∠6 (B), ∠8 (D), ∠7 (C). So vertical angles: ∠5 and ∠7, ∠6 and ∠8.

Now let's re - evaluate the options:
- Option E: ∠6 and ∠8: ∠6 (B at bottom intersection) and ∠8 (D at bottom intersection) are vertical. Correct.
- Option F: ∠5 and ∠7: ∠5 (A at bottom intersection) and ∠7 (C at bottom intersection) are vertical. Correct.
- Option J: ∠2 and ∠4: ∠2 (B at top intersection) and ∠4 (D at top intersection) are vertical. Correct.

Wait, maybe the original diagram's angle numbering: Let's assume the transversal crosses the two horizontal lines. So top horizontal line: intersected by transversal, angles 1 (top - left), 2 (top - right), 4 (bottom - left), 3 (bottom - right). Bottom horizontal line: intersected by transversal, angles 5 (top - left), 6 (top - right), 8 (bottom - left), 7 (bottom - right).

So vertical angle pairs:
1. ∠1 and ∠3? No, wait, ∠1 (top - left of top intersection) and ∠3 (bottom - right of top intersection) are vertical? Wait, no, if you have two lines (horizontal and transversal) intersecting, the vertical angles are opposite. So ∠1 (top - left) and ∠3 (bottom - right) are vertical? Or ∠2 (top - right) and ∠4 (bottom - left) are vertical. Then ∠5 (top - left of bottom intersection) and ∠7 (bottom - right of bottom intersection) are vertical. ∠6 (top - right of bottom intersection) and ∠8 (bottom - left of bottom intersection) are vertical. Also, ∠2 (top - right of top intersection) and ∠4 (bottom - left of top intersection) are vertical. ∠1 (top - left of top intersection) and ∠3 (bottom - right of top intersection) are vertical? Wait, maybe I messed up the angle numbering.

Let's start over. When two lines intersect, they form two pairs of vertical angles. So at the top intersection (transversal and upper horizontal line):
- Angles around the intersection: ∠1 (above transversal, left of horizontal), ∠2 (above transversal, right of horizontal), ∠3 (below transversal, right of horizontal), ∠4 (below transversal, left of horizontal). So vertical angles: ∠1 and ∠3 (opposite), ∠2 and ∠4 (opposite).

At the bottom intersection (transversal and lower horizontal line):
- Angles: ∠5 (above transversal, left of horizontal), ∠6 (above transversal, right of horizontal), ∠7 (below transversal, right of horizontal), ∠8 (below transversal, left of horizontal). So vertical angles: ∠5 and ∠7 (opposite), ∠6 and ∠8 (opposite).

Now let's check the options:
- Option E: 6 & 8: ∠6 (above transversal, right of lower horizontal) and ∠8 (below transversal, left of lower horizontal) are vertical. Correct.
- Option F: 5 & 7: ∠5 (above transversal, left of lower horizontal) and ∠7 (below transversal, right of lower horizontal) are vertical. Correct.
- Option J: 2 & 4: ∠2 (above transversal, right of upper horizontal) and ∠4 (below transversal, left of upper horizontal) are vertical. Correct.

Wait, but maybe the question is to find all correct pairs? But the options are single choices? Wait, no, the problem says "Name each pair of Vertical Angles" and the options are A - J. So we need to find which of the options are vertical angle pairs.

Let's check each option:
- A: 1 & 3: Are they vertical? ∠1 (top - left of top intersection) and ∠3 (bottom - right of top intersection) – yes, they are vertical. Wait, I made a mistake earlier. So ∠1 and ∠3 are vertical.
- B: 1 & 2: Adjacent, linear pair.
- C: 1 & 4: Adjacent.
- D: 1 & 5: Not vertical (different intersections).
- E: 6 & 8: Vertical (as per bottom intersection).
- F: 5 & 7: Vertical (as per bottom intersection).
- G: 7 & 8: Adjacent.
- H: 2 & 8: Not vertical.
- I: 3 & 4: Adjacent.
- J: 2 & 4: Vertical (as per top intersection, ∠2 (top - right) and ∠4 (bottom - left) are vertical).

Wait, now I'm confused. Let's use the definition: Vertical angles are formed by two intersecting lines, and they are non - adjacent, opposite angles.

So at the top intersection (two lines: horizontal and transversal):
- Intersecting lines: horizontal (top) and transversal.
- Angles: ∠1 (between transversal left and horizontal top), ∠2 (between transversal right and horizontal top), ∠4 (between transversal left and horizontal bottom), ∠3 (between transversal right and horizontal bottom).

So vertical angles:
- ∠1 and ∠3 (opposite, non - adjacent)
- ∠2 and ∠4 (opposite, non - adjacent)

At the bottom intersection (two lines: horizontal (bottom) and transversal):
- Angles: ∠5 (between transversal left and horizontal top), ∠6 (between transversal right and horizontal top), ∠8 (between transversal left and horizontal bottom), ∠7 (between transversal right and horizontal bottom).

Vertical angles:
- ∠5 and ∠7 (opposite, non - adjacent)
- ∠6 and ∠8 (opposite, non - adjacent)

Now let's re - check options:
- A: 1 & 3: Vertical (correct)
- E: 6 & 8: Vertical (correct)
- F: 5 & 7: Vertical (correct)
- J: 2 & 4: Vertical (correct)

But maybe the diagram has the angle numbering such that at the top intersection, ∠1 (top - right), ∠2 (top - left)? No, the arrow on the transversal: the transversal is going up to the right. So the top intersection: ∠1 is top - left, ∠2 is top - right, ∠4 is bottom - left, ∠3 is bottom - right.

So ∠1 and ∠3: vertical. ∠2 and ∠4: vertical.

At the bottom intersection: ∠5 is top - left, ∠6 is top - right, ∠8 is bottom - left, ∠7 is bottom - right. So ∠5 and ∠7: vertical. ∠6 and ∠8: vertical.

Now, let's check the options again:
- A: 1 & 3: Vertical (correct)
- E: 6 & 8: Vertical (correct)
- F: 5 & 7: Vertical (correct)
- J: 2 & 4: Vertical (correct)

But maybe the question has multiple correct answers, but let's see the options. Wait, maybe I misread the options. Let's assume the correct vertical angle pairs are:

From the top intersection: ∠2 and ∠4 (J), ∠1 and ∠3 (A)

From the bottom intersection: ∠5 and ∠7 (F), ∠6 and ∠8 (E)

But let's check the original problem again. The problem says "Name each pair of Vertical Angles" and the options are A - J. So we need to find which of the options are vertical angle pairs.

Let's take option J: 2 & 4. ∠2 (top - right of top intersection) and ∠4 (bottom - left of top intersection) are vertical (opposite angles from intersecting lines). So J is correct.

Option F: 5 & 7. ∠5 (top - left of bottom intersection) and ∠7 (bottom - right of bottom intersection) are vertical. Correct.

Option E: 6 & 8. ∠6 (top - right of bottom intersection) and ∠8 (bottom - left of bottom intersection) are vertical. Correct.

Option A: 1 & 3. ∠1 (top - left of top intersection) and ∠3 (bottom - right of top intersection) are vertical. Correct.

But maybe the intended answer is J (2 & 4), F (5 & 7), E (6 & 8), A (1 & 3). But let's check the options again.

Wait, maybe the user made a typo, but let's proceed. Let's assume that among the options, the correct vertical angle pairs are:

- E: 6 & 8 (vertical)
- F: 5 & 7 (vertical)
- J: 2 & 4 (vertical)
- A: 1 & 3 (vertical)

But let's check with the definition again. Vertical angles are equal and opposite. So for example, ∠2 and ∠4: if we measure them, they should be equal. Similarly for others.

So, for example, option J: 2 & 4 are vertical angles (correct).

Answer:

J. 2 & 4, E. 6 & 8, F. 5 & 7, A. 1 & 3 (But if we have to choose one, maybe J as a common vertical angle pair at the top intersection. Wait, maybe the diagram's angle numbering is such that ∠2 and ∠4 are vertical. So the answer is J. 2 & 4)