Practice worksheet for identifying and calculating vertical angles in geometry.

Let's solve the Vertical Angles worksheet step by step. Since I can't see the image directly, I will interpret and solve the problems based on the text you've provided.

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Part A: Conceptual Questions

#### 1. Which of the following is the vertically opposite angle of ∠AOC?
We are given options:
- a) ∠HOK
- b) ∠KOE
- c) ∠EOD
- d) ∠COE

But without a diagram, we need to infer from standard geometry.

In vertical angles, when two lines intersect, they form two pairs of vertically opposite angles. For example, if lines AB and CD intersect at point O, then:
- ∠AOC and ∠BOD are vertically opposite.
- ∠AOD and ∠BOC are vertically opposite.

Assuming the figure shows lines intersecting at point O with points labeled around it (like A, C, E, K, H, D, etc.), and assuming that ∠AOC is formed by rays OA and OC, its vertically opposite angle would be the one directly across the intersection — meaning the angle formed by the opposite rays.

If ∠AOC is between rays OA and OC, then the vertically opposite angle would be between the opposite rays (e.g., OE and OK), depending on labeling.

But let’s suppose the configuration is such that:
- Line AE intersects line CK at point O.
- So, ∠AOC is formed by ray OA and OC.
- The vertically opposite angle would be ∠EOk (if E and K are on the opposite sides).

Looking at the choices:
- ∠HOK → likely not opposite
- ∠KOE → could be
- ∠EOD → possibly
- ∠COE → adjacent

Wait — vertically opposite angles are non-adjacent angles formed by two intersecting lines.

So if ∠AOC is one angle, the vertically opposite angle must be the one directly across — so if A and E are on one line, and C and K on another, then ∠AOC and ∠EOK might be opposite.

But unless we have exact labels, this is tricky.

However, in many such diagrams:
- If lines AC and EK intersect at O,
- Then ∠AOC and ∠EOK are vertically opposite.

So if ∠AOC is formed by rays OA and OC, then the opposite angle is formed by rays OE and OK → ∠EOK.

But option b) ∠KOE is the same as ∠EOK (just reversed).

So Answer: b) ∠KOE

✔ Answer: b) ∠KOE

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#### 2. Identify the vertically opposite angle of ∠EOF from the following:
Options:
- a) ∠BOC
- b) ∠AOC
- c) ∠COD
- d) ∠COE

Again, assume lines intersect at O. ∠EOF is an angle at point O.

The vertically opposite angle to ∠EOF would be the angle directly across — formed by the opposite rays.

Suppose:
- Ray EO and FO form ∠EOF.
- Then the opposite rays would be, say, CO and DO (if E and C are opposite, F and D are opposite).

So if ∠EOF is between EO and FO, then vertically opposite is between CO and DO → ∠COD.

So ∠COD is vertically opposite to ∠EOF.

✔ Answer: c) ∠COD

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#### 3. Write the vertically opposite angle of ∠COD.

From above, if ∠COD is formed by rays CO and DO, then the opposite angle would be formed by the opposite rays — say EO and FO → ∠EOF.

So vertically opposite angle of ∠COD is ∠EOF.

✔ Answer: ∠EOF

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#### 4. Write the vertically opposite angle of ∠FOR.

∠FOR is formed by rays FO and RO.

We need the angle directly opposite.

Assuming R is on the same line as E or something.

But since no diagram, we use logic.

Suppose lines intersect at O. If F and R are two rays, then the opposite rays would be, say, E and O (but O is vertex).

Alternatively, maybe lines FR and EC intersect at O.

Then ∠FOR and ∠EOC might be opposite?

But again, without diagram, hard to tell.

But in typical setups:

If ∠FOR is between F and R, and the other two rays are E and C, then vertically opposite angle is ∠EOC.

But perhaps it's ∠EOC or ∠COE.

But looking back at previous questions, likely:

- ∠FOR → opposite is ∠EOC or ∠COE.

But since ∠EOF was opposite to ∠COD, perhaps symmetry suggests:

If ∠FOR is formed by FO and RO, and if RO is same as CO, then ∠FOR = ∠FOC?

Wait — perhaps better to guess based on common patterns.

Alternatively, maybe the answer is ∠EOC or ∠COE.

But let's suppose the correct pair is ∠FOR and ∠EOC.

But without clear labeling, best to assume based on standard naming.

Wait — maybe from earlier, ∠EOF and ∠COD are opposite.

So if ∠FOR is near ∠EOF, perhaps it's adjacent.

Alternatively, if lines FO and RO intersect with others...

Actually, this is ambiguous without diagram.

But let’s move to Part B, which has diagrams.

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Part B: Find the unknown angles

These are classic vertical angle problems.

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#### 1.
```
A
/ \
/ \
M/ \N
/ \
/ \
O-----------P
\ /
\ /
\ /
B
```

Wait — actually, from the description:

It says:
> m∠1 = 115°
> m∠2 = ___

And it's a pair of intersecting lines forming four angles.

Typical diagram: two lines crossing at a point, forming four angles.

Angles opposite each other are equal (vertical angles).

Adjacent angles are supplementary (add to 180°).

So if ∠1 = 115°, then:

- Its vertical opposite angle is also 115°.
- Adjacent angles are 180° - 115° = 65°.

Now, if ∠1 and ∠2 are adjacent, then ∠2 = 65°.

But we need to know how ∠1 and ∠2 are labeled.

Usually, in such diagrams:
- ∠1 and ∠2 are adjacent.
- So if ∠1 = 115°, then ∠2 = 180° - 115° = 65°.

✔ Answer: m∠2 = 65°

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#### 2.
> m∠1 = 65°
> m∠2 = ___

Same idea.

∠1 = 65°, so adjacent angle ∠2 = 180° - 65° = 115°

(Unless ∠2 is vertical to ∠1, but usually in these diagrams, ∠1 and ∠2 are adjacent.)

But look: the problem says "find the unknown angle", and typically, ∠1 and ∠2 are adjacent.

So:

- If ∠1 = 65°, then ∠2 = 180° - 65° = 115°

✔ Answer: m∠2 = 115°

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#### 3.
> m∠2 = 47°
> m∠3 = ___

Now, ∠2 and ∠3 are adjacent angles formed by intersecting lines.

So if ∠2 = 47°, then ∠3 = 180° - 47° = 133°

(Unless ∠3 is vertical to ∠2, but usually labeled sequentially.)

But if ∠2 and ∠3 are adjacent, then yes.

So:

✔ Answer: m∠3 = 133°

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#### 4.
> m∠1 = 113°
> m∠4 = ___

Again, two intersecting lines.

∠1 and ∠4 are likely adjacent or vertical.

But in most such diagrams:
- ∠1 and ∠4 are adjacent angles.

So if ∠1 = 113°, then ∠4 = 180° - 113° = 67°

Alternatively, if ∠1 and ∠4 are vertical, they’d be equal, but 113° ≠ 67°, so not equal.

So they are adjacent.

✔ Answer: m∠4 = 67°

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✔ Final Answers:

#### Part A:
1. b) ∠KOE
2. c) ∠COD
3. ∠EOF
4. ∠EOC (or similar; depends on diagram, but likely ∠EOC)

But for #4, if ∠FOR is formed by FO and RO, and if the opposite rays are EO and CO, then vertically opposite angle is ∠EOC.

So Answer: ∠EOC

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#### Part B:
1. m∠2 = 65°
2. m∠2 = 115°
3. m∠3 = 133°
4. m∠4 = 67°

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🔍 Explanation Summary:

- Vertical angles are opposite angles formed when two lines intersect. They are equal.
- Adjacent angles are next to each other and add up to 180° (supplementary).
- Use these two rules to solve all problems.

For example:
- In Part B, if one angle is 115°, its adjacent angle is 65°.
- Its vertical opposite is also 115°.

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Let me know if you'd like a visual explanation or help interpreting the diagram!