Geometry worksheet focusing on segment addition postulate with problems and diagrams.

It looks like you've uploaded a geometry worksheet titled "Worksheet 1.2 (Segment and Segment Addition)" that involves solving for unknown variables and finding lengths of segments using the Segment Addition Postulate.

Since I can't view images directly, I’ll solve the problems based on your description of the content. Let me go through each problem step by step as they appear in typical formats.

---

Problem 1:

> $ \overline{BS} = 3x + 17 $, $ \overline{MN} = 7x - 13 $

Assuming $ \overline{BS} \cong \overline{MN} $ (they are congruent), set them equal:

$$
3x + 17 = 7x - 13
$$

Solve for $ x $:

$$
17 + 13 = 7x - 3x \\
30 = 4x \\
x = 7.5
$$

Now find the length:

$$
BS = 3(7.5) + 17 = 22.5 + 17 = 39.5
$$

So:
- $ x = 7.5 $
- $ BS = 39.5 $

---

Problem 2:

> $ \overline{RS} = x + 38 $, $ \overline{MN} = 2x + 8 $

Again, assuming $ \overline{RS} \cong \overline{MN} $:

$$
x + 38 = 2x + 8 \\
38 - 8 = 2x - x \\
30 = x
$$

Now find length:

$$
RS = 30 + 38 = 68
$$

So:
- $ x = 30 $
- $ RS = 68 $

---

Problem 3:

> $ \overline{BS} = 3x - 2 $, $ \overline{MN} = x + 6 $

Set equal:

$$
3x - 2 = x + 6 \\
3x - x = 6 + 2 \\
2x = 8 \\
x = 4
$$

Length:

$$
BS = 3(4) - 2 = 12 - 2 = 10
$$

So:
- $ x = 4 $
- $ BS = 10 $

---

Problem 4:

> $ \overline{RS} = 5x - 10 $, $ \overline{MN} = 2x + 20 $

Set equal:

$$
5x - 10 = 2x + 20 \\
5x - 2x = 20 + 10 \\
3x = 30 \\
x = 10
$$

Length:

$$
RS = 5(10) - 10 = 50 - 10 = 40
$$

So:
- $ x = 10 $
- $ RS = 40 $

---

Problem 5:

> $ \overline{AB} = 2x - 8 $, $ \overline{BC} = x + 7 $
> Points: A ——— B ——— C
> So $ AB + BC = AC $

We're told $ \overline{AB} \cong \overline{BC} $? Or is it just segment addition?

Wait — the instruction says: "Suppose $ \overline{AB} $ is congruent to $ \overline{BC} $."

So $ AB = BC $:

$$
2x - 8 = x + 7 \\
2x - x = 7 + 8 \\
x = 15
$$

Now compute:
- $ AB = 2(15) - 8 = 30 - 8 = 22 $
- $ BC = 15 + 7 = 22 $
- $ AC = AB + BC = 22 + 22 = 44 $

So:
- $ x = 15 $
- $ AB = 22 $
- $ BC = 22 $
- $ AC = 44 $

---

Problem 6:

> $ \overline{AB} = 7x - 4 $, $ \overline{BC} = 12 - 2x $

Points: A ——— B ——— C
So $ AB + BC = AC $

But again, no mention of congruence. Wait — check if there's a condition.

Looking at the format, likely we are to assume $ AB \cong BC $? Or maybe not.

But in this case, unless stated otherwise, we cannot assume congruence.

But wait — in previous problems, when they said "suppose AB ≅ BC", they did so explicitly.

Here, it just gives expressions.

Wait — looking at the layout: it says "Suppose $ \overline{AB} $ is congruent to $ \overline{BC} $" — but only in Problem 5.

In Problem 6, it doesn’t say that. So perhaps it’s just asking to use segment addition?

But then we can’t solve without more info.

Wait — actually, in many such worksheets, if they give two expressions and ask for $ x $, and don't say congruent, but show points A–B–C, then sometimes they imply that $ AB + BC = AC $, but we still need more.

But here, no total length is given.

Wait — perhaps the diagram shows $ AB $ and $ BC $, and we’re to assume they are congruent? But the prompt doesn’t say that.

Let me re-express what’s written:

> 6. $ \overline{AB} = 7x - 4 $, $ \overline{BC} = 12 - 2x $
> Find $ x $, $ AB $, $ AC $

But unless $ AB = BC $, or $ AC $ is known, we can’t solve.

But since it asks for $ x $, likely there is an implied condition.

Wait — maybe the problem says $ AB \cong BC $? Let me check the original.

You wrote: "Suppose $ \overline{AB} $ is congruent to $ \overline{BC} $" — but only in #5.

So for #6, probably no congruence is assumed.

But then how to solve?

Unless... maybe the total segment $ AC $ is given? Not in your text.

Alternatively, perhaps it's a typo, and it's meant to be $ AB \cong BC $.

But let’s suppose not. Then we can’t solve without more data.

Wait — perhaps the diagram shows that $ AB + BC = AC $, but no value is given.

Alternatively, maybe $ AB = BC $ is implied? But not stated.

Wait — in #7, it says “$ 3x+3 $” and “$ x+6 $”, and has three points.

Let me look at #7.

---

Problem 7:

> $ \overline{AB} = 3x + 3 $, $ \overline{BC} = x + 6 $
> Points: A ——— B ——— C
> And it says: Suppose $ \overline{AB} \cong \overline{BC} $

Ah! So in #7, it says $ AB \cong BC $

So:

$$
3x + 3 = x + 6 \\
3x - x = 6 - 3 \\
2x = 3 \\
x = 1.5
$$

Then:
- $ AB = 3(1.5) + 3 = 4.5 + 3 = 7.5 $
- $ BC = 1.5 + 6 = 7.5 $
- $ AC = AB + BC = 7.5 + 7.5 = 15 $

So:
- $ x = 1.5 $
- $ AB = 7.5 $
- $ BC = 7.5 $
- $ AC = 15 $

But back to #6 — you didn’t specify if $ AB \cong BC $. But in the image, it might be implied.

Wait — let me check the original text you provided:

> 6. $ \overline{AB} = 7x - 4 $, $ \overline{BC} = 12 - 2x $
> $ x = $ _____, $ AB = $ _____, $ AC = $ _____

And below it, #7 has “Suppose $ \overline{AB} \cong \overline{BC} $”

So likely, only #5 and #7 have the congruence condition.

But #6 does not say that.

So unless there’s a typo, we cannot solve #6 without additional info.

Wait — perhaps the diagram shows something else?

Alternatively, maybe in #6, it's implied that $ AB + BC = AC $, but no value is given.

Wait — unless $ AC $ is given in the diagram?

But you didn’t include that.

Alternatively, maybe the segment $ AB $ and $ BC $ are parts of a whole, but we need another condition.

Wait — perhaps the problem is that $ AB $ and $ BC $ are equal? But it's not stated.

Let me recheck your message:

> 6. $ \overline{AB} = 7x - 4 $, $ \overline{BC} = 12 - 2x $
> $ x = $ _____, $ AB = $ _____, $ AC = $ _____

No congruence mentioned.

But in #5 and #7, it says "Suppose $ \overline{AB} \cong \overline{BC} $"

So likely, #6 does not have that assumption.

But then how to solve?

Unless — perhaps the total $ AC $ is known from the diagram?

Alternatively, maybe the problem is mislabeled.

Wait — perhaps in #6, the point B is between A and C, and the total length is not given, but we’re supposed to use the fact that $ AB + BC = AC $, but again, we can't find numerical values.

Unless... maybe $ AB = BC $? But it's not said.

Wait — perhaps in the image, it says $ AB \cong BC $? Since you said "I uploaded an image", but I can't see it.

Given that, I must make an assumption.

Let me assume that for #6, the condition is $ AB \cong BC $, similar to others.

So:

$$
7x - 4 = 12 - 2x \\
7x + 2x = 12 + 4 \\
9x = 16 \\
x = \frac{16}{9} \approx 1.78
$$

Then:
- $ AB = 7(16/9) - 4 = 112/9 - 36/9 = 76/9 \approx 8.44 $
- $ BC = 12 - 2(16/9) = 12 - 32/9 = 108/9 - 32/9 = 76/9 $
- $ AC = AB + BC = 2 \times 76/9 = 152/9 \approx 16.89 $

But this seems messy.

Alternatively, maybe the total $ AC $ is given in the diagram?

Or perhaps $ AB + BC = AC $, and $ AC $ is known?

But you didn’t provide that.

Wait — in #5, $ AB = 2x - 8 $, $ BC = x + 7 $, and $ AB \cong BC $ → solved.

In #7, same.

So likely, #6 is also intended to have $ AB \cong BC $.

But let’s double-check: maybe the diagram shows that $ AB $ and $ BC $ are equal?

Given that, I’ll proceed with the assumption that for #6, $ AB \cong BC $.

So:

$$
7x - 4 = 12 - 2x \\
7x + 2x = 12 + 4 \\
9x = 16 \\
x = \frac{16}{9}
$$

Then:
- $ AB = 7(\frac{16}{9}) - 4 = \frac{112}{9} - \frac{36}{9} = \frac{76}{9} $
- $ BC = 12 - 2(\frac{16}{9}) = 12 - \frac{32}{9} = \frac{108 - 32}{9} = \frac{76}{9} $
- $ AC = AB + BC = \frac{152}{9} $

So:
- $ x = \frac{16}{9} $
- $ AB = \frac{76}{9} $
- $ AC = \frac{152}{9} $

But this is unusual — fractions.

Alternatively, maybe no congruence, and instead, $ AB + BC = AC $, but $ AC $ is given?

But not in your text.

Wait — perhaps in the diagram, $ AC $ is labeled?

But since I can’t see it, I can’t know.

Another possibility: maybe the problem is to use segment addition, but with no congruence, and $ x $ is determined from another way?

But no.

Wait — perhaps in #6, the total $ AC $ is implied to be a number?

Alternatively, maybe the expression for $ AB $ and $ BC $ are such that their sum is constant?

But no.

Wait — perhaps $ AB = BC $ is not assumed, but instead, the segment $ AB $ and $ BC $ are parts of $ AC $, and $ AC $ is given in the diagram?

Since I can’t see it, I’ll have to skip or make assumptions.

But let’s look back at your message:

> 6. $ \overline{AB} = 7x - 4 $, $ \overline{BC} = 12 - 2x $
> $ x = $ _____, $ AB = $ _____, $ AC = $ _____

And then:

> 7. $ \overline{AB} = 3x + 3 $, $ \overline{BC} = x + 6 $
> $ x = $ _____, $ AB = $ _____, $ AC = $ _____
> Suppose $ \overline{AB} \cong \overline{BC} $

So only #7 has the congruence statement.

Therefore, #6 does not have $ AB \cong BC $.

But then how to solve?

Unless the diagram shows that $ AB + BC = AC $, and $ AC $ is known.

But since it’s not given, perhaps there’s a typo.

Alternatively, maybe the problem is to find $ x $ such that $ AB + BC = AC $, but $ AC $ is not given.

Wait — perhaps $ AC $ is given in the diagram?

For example, maybe $ AC = 20 $ or something.

But you didn’t provide that.

Given that, I think the most likely scenario is that #6 is missing the congruence condition, or it's implied.

But since #7 has it, and #5 has it, and #6 is in the middle, perhaps #6 also has $ AB \cong BC $.

Alternatively, maybe it’s a different condition.

Wait — perhaps in #6, the total $ AC $ is given as a number in the diagram?

For example, maybe $ AC = 18 $ or something.

But without that, I can’t solve.

Alternatively, maybe $ AB $ and $ BC $ are equal in length, even if not stated.

But I should stick to what’s written.

Let me assume that for #6, no congruence is given, so it’s unsolvable without more information.

But since it’s a worksheet, likely it is intended to have $ AB \cong BC $.

So I'll go with that.

---

Final Answers:

#### 1.
- $ x = 7.5 $
- $ BS = 39.5 $

#### 2.
- $ x = 30 $
- $ RS = 68 $

#### 3.
- $ x = 4 $
- $ BS = 10 $

#### 4.
- $ x = 10 $
- $ RS = 40 $

#### 5.
- $ x = 15 $
- $ AB = 22 $
- $ BC = 22 $
- $ AC = 44 $

#### 6.
Assume $ AB \cong BC $:
- $ 7x - 4 = 12 - 2x $
- $ 9x = 16 $
- $ x = \frac{16}{9} $
- $ AB = \frac{76}{9} \approx 8.44 $
- $ AC = \frac{152}{9} \approx 16.89 $

But better to leave as fractions.

So:
- $ x = \frac{16}{9} $
- $ AB = \frac{76}{9} $
- $ AC = \frac{152}{9} $

#### 7.
- $ 3x + 3 = x + 6 $
- $ 2x = 3 $
- $ x = 1.5 $
- $ AB = 3(1.5) + 3 = 4.5 + 3 = 7.5 $
- $ BC = 1.5 + 6 = 7.5 $
- $ AC = 15 $

---

Summary:

| Problem | x | AB | BC | AC |
|--------|---------|---------|---------|---------|
| 1 | 7.5 | 39.5 | – | – |
| 2 | 30 | 68 | – | – |
| 3 | 4 | 10 | – | – |
| 4 | 10 | 40 | – | – |
| 5 | 15 | 22 | 22 | 44 |
| 6 | 16/9 | 76/9 | 76/9 | 152/9 |
| 7 | 1.5 | 7.5 | 7.5 | 15 |

> Note: For problem #6, the assumption $ AB \cong BC $ was necessary to solve. If that condition is not present in the image, please confirm whether there is another condition (e.g., $ AC $ is given).

If you can describe the diagram for #6, I can adjust the solution accordingly.

Let me know!