Figure 21 displays six different geometric scenarios where students must calculate the value of angle x using angle properties.

Let's solve each of the six angle problems in Fig 21 step by step. These are all geometry problems involving angles at a point, straight lines, and vertically opposite angles.

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(i)

Given:
- A straight line AC with point O on it.
- ∠AOB = 60°
- ∠BOC = x°

Since AO and OC form a straight line, the total angle is 180°.

So:
$$
∠AOB + ∠BOC = 180^\circ \\
60^\circ + x^\circ = 180^\circ \\
x = 180 - 60 = 120^\circ
$$

✔ Answer: x = 120°

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(ii)

Given:
- A straight line PR with point O on it.
- ∠POQ = 3x°
- ∠QOR = 2x°

These two angles make up a straight line:

$$
3x + 2x = 180^\circ \\
5x = 180^\circ \\
x = \frac{180}{5} = 36^\circ
$$

✔ Answer: x = 36°

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(iii)

Given:
- A straight line LM with point O on it.
- ∠POL = 35°
- ∠MON = 60°
- ∠PON = x°

Note: Points P, O, N form an angle x° between them.

But we need to consider that ∠POL + ∠PON + ∠NOM = 180° (since they lie on a straight line).

Wait — actually, from the diagram:

- From left to right: L → O → M is a straight line.
- Ray OP makes 35° with LO.
- Ray ON makes 60° with OM.
- So, the angle between OP and ON is x°.

So the full straight line is:
$$
∠LOP + ∠PON + ∠NOM = 180^\circ \\
35^\circ + x^\circ + 60^\circ = 180^\circ \\
x = 180 - 95 = 85^\circ
$$

✔ Answer: x = 85°

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(iv)

This is a point O where multiple rays meet.

We are given:
- ∠COA = 83°
- ∠AOB = 47°
- ∠BOE = 75°
- ∠DOE = 92°
- We are to find x°, which is ∠COB?

Wait — looking carefully: The angle marked as x° is ∠COB.

Let’s label the angles around point O.

From the diagram:
- ∠COA = 83°
- ∠AOB = 47°
- So ∠COB = ∠COA + ∠AOB = 83° + 47° = 130°

But wait — are these adjacent? Yes, if C-O-A-O-B are in order.

But let's check the full circle.

We can also use the fact that sum of angles around a point is 360°.

List known angles:
- ∠COD = ?
- ∠DOE = 92°
- ∠EOB = 75°
- ∠BOA = 47°
- ∠AOC = 83°

Wait — but ∠COA = 83°, so from C to A is 83°.

Now, going around the point O:

Let’s go clockwise:
1. ∠COA = 83°
2. ∠AOB = 47°
3. ∠BOE = 75°
4. ∠EOD = 92°
5. ∠DOC = ?

Wait — but we don’t know ∠DOC yet.

But we are told that ∠COB = x°

From earlier: ∠COB = ∠COA + ∠AOB = 83° + 47° = 130°

So x = 130°

But let’s verify this with full circle:

Sum of all angles around point O should be 360°.

Let’s compute:

- ∠COA = 83°
- ∠AOB = 47°
- ∠BOE = 75°
- ∠EOD = 92°
- ∠DOC = ? → But we don’t have D to C directly.

Wait — maybe the angle between D and C is missing.

Alternatively, perhaps ∠DOC is not needed.

But if we assume the rays are in order: C, A, B, E, D, back to C.

Then:

- ∠COA = 83°
- ∠AOB = 47° → total so far: 130°
- ∠BOE = 75° → total: 205°
- ∠EOD = 92° → total: 297°
- Then ∠DOC = 360° - 297° = 63°

But we were asked for ∠COB, which is ∠COA + ∠AOB = 83 + 47 = 130°

✔ Answer: x = 130°

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(v)

At point O, four rays: R, Q, P, S.

Angles given:
- ∠ROQ = 2x°
- ∠QOP = x°
- ∠POS = 2x°
- ∠SOR = 3x°

All these angles go around point O.

So sum of angles around a point = 360°

Add them:
$$
2x + x + 2x + 3x = 8x = 360^\circ \\
x = \frac{360}{8} = 45^\circ
$$

✔ Answer: x = 45°

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(vi)

Two intersecting lines forming an "X" shape.

One angle is 105°, another is 3x°.

They are vertically opposite angles, or adjacent?

Looking at the diagram:

- The angle marked 105° and 3x° are adjacent angles on a straight line (they form a linear pair).

So:
$$
105^\circ + 3x^\circ = 180^\circ \\
3x = 180 - 105 = 75 \\
x = 25^\circ
$$

Alternatively, if they were vertically opposite, they would be equal, but 105 ≠ 3x unless x=35, but then 3x=105, so that could be.

Wait — let’s look carefully.

In figure (vi), two lines cross at O.

One angle is labeled 105°, and the opposite one is labeled 3x°.

If they are vertically opposite, then:
$$
3x = 105 \Rightarrow x = 35^\circ
$$

But the angle next to 105° is 3x°, and since they are adjacent on a straight line, they must add to 180°.

So:
$$
105 + 3x = 180 \Rightarrow 3x = 75 \Rightarrow x = 25
$$

But then the opposite angle would be 3x = 75°, and the other vertical angle would be 105°, which matches.

So yes, 105° and 3x° are adjacent, forming a straight line.

Therefore:
$$
3x = 180 - 105 = 75 \Rightarrow x = 25^\circ
$$

✔ Answer: x = 25°

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✔ Final Answers:

| Figure | Value of x |
|--------|------------|
| (i) | 120° |
| (ii) | 36° |
| (iii) | 85° |
| (iv) | 130° |
| (v) | 45° |
| (vi) | 25° |

Let me know if you'd like diagrams or further explanation!