Calculate the missing angles on a straight line and around a point.

Let's solve each of the angle problems step by step. The goal is to calculate the missing angles using the properties of angles on a straight line (sum = 180°), angles around a point (sum = 360°), and right angles (90°).

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Row 1: Angles on a Straight Line (Sum = 180°)

#### 1. First diagram:
- Given: 70°, find angle a
- Since they are on a straight line:
$$
a = 180^\circ - 70^\circ = 110^\circ
$$

✔ a = 110°

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#### 2. Second diagram:
- Given: 100°, find angle b
$$
b = 180^\circ - 100^\circ = 80^\circ
$$

✔ b = 80°

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#### 3. Third diagram:
- Given: 40°, find angle c
$$
c = 180^\circ - 40^\circ = 140^\circ
$$

✔ c = 140°

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Row 2: Angles on a Straight Line (with multiple angles)

#### 4. Fourth diagram:
- Given: 50°, 90°, find a
- All three angles form a straight line:
$$
50^\circ + 90^\circ + a = 180^\circ \\
a = 180^\circ - 140^\circ = 40^\circ
$$

✔ a = 40°

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#### 5. Fifth diagram:
- Given: 30°, 50°, find b
- On a straight line:
$$
30^\circ + b + 50^\circ = 180^\circ \\
b = 180^\circ - 80^\circ = 100^\circ
$$

✔ b = 100°

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#### 6. Sixth diagram:
- Given: 80°, 30°, find c
- These two angles are adjacent on a straight line:
$$
c = 180^\circ - (80^\circ + 30^\circ) = 180^\circ - 110^\circ = 70^\circ
$$

✔ c = 70°

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Row 3: Intersecting Lines & Right Angles

#### 7. Seventh diagram:
- Given: 40°, and a right angle (90°), find a
- This forms a straight line with three angles: 40°, 90°, and a
$$
40^\circ + 90^\circ + a = 180^\circ \\
a = 180^\circ - 130^\circ = 50^\circ
$$

✔ a = 50°

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#### 8. Eighth diagram:
- Given: 120°, find b
- There’s a right angle (90°) marked at the intersection.
- The angles around the point must sum to 360°, but here we see that b is opposite or adjacent?

Wait — let's analyze carefully.

We have a horizontal line, vertical line, and one diagonal ray forming an angle of 120° with the horizontal.

But there's a right angle (90°) symbol at the corner where the vertical and horizontal meet.

So, from the horizontal line, going up vertically is 90°, and then from vertical to the ray is part of the 120°.

Actually, the 120° is between the horizontal and the diagonal ray.

So, the angle b is the remaining angle on the straight line.

Wait — no. Let's look again.

It appears that:

- A horizontal line.
- A vertical line upward (forming a 90° angle).
- A diagonal line going down-right, making a 120° angle with the horizontal on the other side.

But b is the angle between the vertical and the diagonal line.

Let’s think differently.

The total around the point is 360°.

We have:
- 120° (between horizontal and diagonal)
- 90° (vertical and horizontal)
- So the remaining angle b is between the vertical and the diagonal.

But wait — actually, the 120° is not on the same side as the right angle.

Looking closely: The 120° is shown below the horizontal line, while the right angle is above.

So the angles around the point:

- 120° (bottom-left)
- 90° (top-left, between horizontal and vertical)
- Then b is the angle between vertical and diagonal on the bottom-right?
Wait — this might be better interpreted as:

The horizontal line is split into two parts by the diagonal ray. One angle is 120°, so the other angle on that line is:
$$
180^\circ - 120^\circ = 60^\circ
$$

Now, since the vertical line makes a 90° angle with the horizontal, and the diagonal ray makes a 60° angle with the horizontal, then the angle b between the vertical and the diagonal is:
$$
90^\circ - 60^\circ = 30^\circ
$$

Alternatively, if you go from vertical down to diagonal: it's 90° minus 60° = 30°.

But b is labeled in the space between the vertical and the diagonal — yes.

✔ b = 30°

Wait — double-check.

Actually, let's use the fact that angles on a straight line add to 180°.

The 120° is on one side of the horizontal line. The angle adjacent to it (on the same line) is 60°.

That 60° is between the diagonal and the horizontal.

The vertical line is perpendicular to the horizontal → 90°.

So the angle between the diagonal and the vertical is:
$$
90^\circ - 60^\circ = 30^\circ
$$

✔ b = 30°

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#### 9. Ninth diagram:
- Given: 130°, find c
- This is a straight line intersected by a vertical line.
- The 130° is one angle formed between the diagonal and the horizontal.
- The angle c is between the vertical and the diagonal.

Again, use:
- Angle between horizontal and diagonal: 130°
- So the adjacent angle on the same line is: $180^\circ - 130^\circ = 50^\circ$

This 50° is between the diagonal and the horizontal.

Vertical is 90° from horizontal.

So angle between diagonal and vertical is:
$$
90^\circ - 50^\circ = 40^\circ
$$

But c is labeled at the top, between vertical and diagonal — yes.

✔ c = 40°

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Row 4: Angles Around a Point (Total = 360°)

#### 10. Tenth diagram:
- Circle divided into 5 equal sectors, each labeled a
- So:
$$
5a = 360^\circ \\
a = \frac{360^\circ}{5} = 72^\circ
$$

✔ a = 72°

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#### 11. Eleventh diagram:
- Circle with one sector labeled 120°, the rest is b
- So:
$$
b = 360^\circ - 120^\circ = 240^\circ
$$

✔ b = 240°

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#### 12. Twelfth diagram:
- Circle divided into 3 sectors: two labeled c, one labeled 120°
- So:
$$
2c + 120^\circ = 360^\circ \\
2c = 240^\circ \\
c = 120^\circ
$$

✔ c = 120°

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✔ Final Answers:

| Diagram | Missing Angle | Value |
|--------|---------------|-------|
| 1 | a | 110° |
| 2 | b | 80° |
| 3 | c | 140° |
| 4 | a | 40° |
| 5 | b | 100° |
| 6 | c | 70° |
| 7 | a | 50° |
| 8 | b | 30° |
| 9 | c | 40° |
| 10 | a | 72° |
| 11 | b | 240° |
| 12 | c | 120° |

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Summary of Answers:

1. 110°
2. 80°
3. 140°
4. 40°
5. 100°
6. 70°
7. 50°
8. 30°
9. 40°
10. 72°
11. 240°
12. 120°

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