Geometry worksheet focusing on the Segment Addition Postulate with problems involving algebraic expressions and segment lengths.

Let's solve each problem step by step using the Segment Addition Postulate, which states:

> If point $ M $ is between points $ L $ and $ N $, then:
> $$
> LM + MN = LN
> $$

We'll apply this postulate to each question.

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1. If $ LM = 22 $ and $ MN = 15 $, find $ LN $.

Using the Segment Addition Postulate:
$$
LN = LM + MN = 22 + 15 = \boxed{37}
$$

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2. If $ LN = 54 $ and $ LM = 31 $, find $ MN $.

$$
LN = LM + MN \Rightarrow 54 = 31 + MN
\Rightarrow MN = 54 - 31 = \boxed{23}
$$

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3. If $ RT = 36 $, find the value of $ x $.

From the diagram:
- $ RS = 6x + 1 $
- $ ST = x + 7 $
- $ RT = RS + ST $

So:
$$
RT = (6x + 1) + (x + 7) = 7x + 8
$$
Given $ RT = 36 $:
$$
7x + 8 = 36 \\
7x = 28 \\
x = \boxed{4}
$$

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4. If $ DF = 9x - 39 $, find $ EF $.

From the diagram:
- $ DE = 47 $
- $ EF = 3x + 10 $
- $ DF = DE + EF $

So:
$$
DF = 47 + (3x + 10) = 3x + 57
$$
But we're told $ DF = 9x - 39 $, so:
$$
9x - 39 = 3x + 57 \\
9x - 3x = 57 + 39 \\
6x = 96 \\
x = 16
$$

Now plug into $ EF = 3x + 10 $:
$$
EF = 3(16) + 10 = 48 + 10 = \boxed{58}
$$

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5. If $ UW = 6x - 35 $, find $ UW $.

From the diagram:
- $ UV = 19 $
- $ VW = 4x - 20 $
- $ UW = UV + VW $

So:
$$
UW = 19 + (4x - 20) = 4x - 1
$$
But we're told $ UW = 6x - 35 $, so:
$$
6x - 35 = 4x - 1 \\
6x - 4x = -1 + 35 \\
2x = 34 \\
x = 17
$$

Now plug into $ UW = 6x - 35 $:
$$
UW = 6(17) - 35 = 102 - 35 = \boxed{67}
$$

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6. If $ HJ = 7x - 27 $, find the value of $ x $.

From the diagram:
- $ HI = 3x - 5 $
- $ IJ = x - 1 $
- $ HJ = HI + IJ $

So:
$$
HJ = (3x - 5) + (x - 1) = 4x - 6
$$
Set equal to given $ HJ = 7x - 27 $:
$$
4x - 6 = 7x - 27 \\
-6 + 27 = 7x - 4x \\
21 = 3x \\
x = \boxed{7}
$$

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7. If $ BD = 7x - 10 $, $ BC = 4x - 29 $, and $ CD = 5x - 9 $, find each value.

From the diagram: $ B $–$ C $–$ D $, so:
$$
BD = BC + CD
$$

Substitute:
$$
7x - 10 = (4x - 29) + (5x - 9) \\
7x - 10 = 9x - 38 \\
-10 + 38 = 9x - 7x \\
28 = 2x \\
x = \boxed{14}
$$

Now compute each segment:

- $ BC = 4x - 29 = 4(14) - 29 = 56 - 29 = \boxed{27} $
- $ CD = 5x - 9 = 5(14) - 9 = 70 - 9 = \boxed{61} $
- $ BD = 7x - 10 = 7(14) - 10 = 98 - 10 = \boxed{88} $

Check: $ BC + CD = 27 + 61 = 88 = BD $ ✔

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8. If $ \overline{BD} \cong \overline{BC} $, $ BD = 5x - 26 $, $ BC = 2x + 1 $, and $ AC = 43 $, find $ AB $.

Given:
- $ \overline{BD} \cong \overline{BC} \Rightarrow BD = BC $
- So:
$$
5x - 26 = 2x + 1 \\
5x - 2x = 1 + 26 \\
3x = 27 \\
x = 9
$$

Now find $ BC $:
$$
BC = 2x + 1 = 2(9) + 1 = 18 + 1 = 19
$$

Now look at the diagram: Points are $ A $–$ B $–$ C $, and $ B $–$ D $ (perpendicular? Not important for length). But we’re told $ AC = 43 $, and $ AC = AB + BC $

So:
$$
AC = AB + BC \Rightarrow 43 = AB + 19 \\
AB = 43 - 19 = \boxed{24}
$$

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✔ Final Answers:

1. $ LN = \boxed{37} $
2. $ MN = \boxed{23} $
3. $ x = \boxed{4} $
4. $ EF = \boxed{58} $
5. $ UW = \boxed{67} $
6. $ x = \boxed{7} $
7. $ x = \boxed{14},\ BC = \boxed{27},\ CD = \boxed{61},\ BD = \boxed{88} $
8. $ AB = \boxed{24} $

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