** this is a 2-page document! **

use the diagram below to
answer questions 1 and 2.
diagram: l---m---n

1. if ( lm = 22 ) and ( mn = 15 ), find ( ln ).

2. if ( ln = 54 ) and ( lm = 31 ), find ( mn ).

3. if ( rt = 36 ), find the value of ( x ).
diagram: ( r )---( 6x + 1 )---( s )---( x + 7 )---( t )

4. if ( df = 9x - 39 ), find ( ef ).
diagram: ( d )---( 4? )---( e )---( 8x + 10 )---( f )

5. if ( uw = 6x - 35 ), find ( uw ).
diagram: ( u )---( 19 )---( v )---( 4x - 20 )---( w )

6. if ( hj = 7x - 27 ), find the value of ( x ).
diagram: ( h )---( 3x - 5 )---( i )---( x - 1 )---( j )

7. if ( bd = 7x - 10 ), ( bc = 4x - 29 ), and ( cd = 5x - 9 ), find each value.
diagram: ( b )---( c )---( d )
( x = ______ )
( bc = ______ )
( cd = ______ )
( bd = ______ )

8. if ( overline{bd} cong overline{bc} ), ( bd = 5x - 26 ), ( bc = 2x + 1 ), and ( ac = 43 ), find ( ab ).
diagram: ( a )---( b )---( c ), with ( d ) perpendicular to ( b )

Question

** this is a 2-page document! **

use the diagram below to
answer questions 1 and 2.
diagram: l---m---n

1. if ( lm = 22 ) and ( mn = 15 ), find ( ln ).

2. if ( ln = 54 ) and ( lm = 31 ), find ( mn ).

3. if ( rt = 36 ), find the value of ( x ).
diagram: ( r )---( 6x + 1 )---( s )---( x + 7 )---( t )

4. if ( df = 9x - 39 ), find ( ef ).
diagram: ( d )---( 4? )---( e )---( 8x + 10 )---( f )

5. if ( uw = 6x - 35 ), find ( uw ).
diagram: ( u )---( 19 )---( v )---( 4x - 20 )---( w )

6. if ( hj = 7x - 27 ), find the value of ( x ).
diagram: ( h )---( 3x - 5 )---( i )---( x - 1 )---( j )

7. if ( bd = 7x - 10 ), ( bc = 4x - 29 ), and ( cd = 5x - 9 ), find each value.
diagram: ( b )---( c )---( d )
( x = ______ )
( bc = ______ )
( cd = ______ )
( bd = ______ )

8. if ( overline{bd} cong overline{bc} ), ( bd = 5x - 26 ), ( bc = 2x + 1 ), and ( ac = 43 ), find ( ab ).
diagram: ( a )---( b )---( c ), with ( d ) perpendicular to ( b )

Accepted Answer

### Question 1: If $ LM = 22 $ and $ MN = 15 $, find $ LN $.
# Explanation:
## Step 1: Recall segment addition postulate
For collinear points $ L $, $ M $, $ N $, $ LN = LM + MN $.
## Step 2: Substitute values
Substitute $ LM = 22 $ and $ MN = 15 $ into the formula: $ LN = 22 + 15 $.
## Step 3: Calculate
$ 22 + 15 = 37 $.
# Answer: $ 37 $

### Question 2: If $ LN = 54 $ and $ LM = 31 $, find $ MN $.
# Explanation:
## Step 1: Recall segment addition postulate
$ LN = LM + MN $, so $ MN = LN - LM $.
## Step 2: Substitute values
Substitute $ LN = 54 $ and $ LM = 31 $: $ MN = 54 - 31 $.
## Step 3: Calculate
$ 54 - 31 = 23 $.
# Answer: $ 23 $

### Question 3: If $ RT = 36 $, find the value of $ x $. (Diagram: $ R $---$ 6x + 1 $---$ S $---$ x + 7 $---$ T $)
# Explanation:
## Step 1: Segment addition postulate
$ RT = RS + ST $, so $ 36 = (6x + 1) + (x + 7) $.
## Step 2: Simplify equation
Combine like terms: $ 36 = 7x + 8 $.
## Step 3: Solve for $ x $
Subtract 8 from both sides: $ 36 - 8 = 7x $ → $ 28 = 7x $. Divide by 7: $ x = \frac{28}{7} = 4 $.
# Answer: $ 4 $

### Question 4: If $ DE = 47 $ (assuming typo, maybe $ DE = 47 $? Wait, original: $ D $---$ 4? $ (maybe $ 47 $?)---$ E $---$ 8x + 10 $---$ F $, and $ DF = 9x - 39 $. Wait, maybe $ DE = 47 $? Wait, no, maybe $ DE = 47 $ is a typo, but let's assume $ DE = 47 $ (or maybe $ DE = 47 $ is correct). Wait, segment addition: $ DF = DE + EF $, so $ 9x - 39 = 47 + 8x + 10 $? Wait, no, maybe $ DE = 47 $ is $ DE = 47 $, so:
## Step 1: Segment addition
$ DF = DE + EF $ → $ 9x - 39 = 47 + 8x + 10 $? Wait, no, maybe $ DE = 47 $ is $ DE = 47 $, so $ 9x - 39 = 47 + (8x + 10) $? Wait, no, let's re-express. If $ D $---$ DE $---$ E $---$ EF $---$ F $, then $ DF = DE + EF $. So $ 9x - 39 = DE + 8x + 10 $. Wait, maybe $ DE = 47 $ (the "4?" is a typo, maybe 47). So:
## Step 1: Substitute into segment addition
$ 9x - 39 = 47 + 8x + 10 $.
## Step 2: Simplify
$ 9x - 39 = 8x + 57 $.
## Step 3: Solve for $ x $
Subtract $ 8x $: $ x - 39 = 57 $. Add 39: $ x = 96 $. Then $ EF = 8x + 10 = 8(96) + 10 = 768 + 10 = 778 $. Wait, that seems large. Maybe $ DE = 4 $ (the "4?" is 4). Let's try that:
## Step 1: Segment addition with $ DE = 4 $
$ 9x - 39 = 4 + 8x + 10 $.
## Step 2: Simplify
$ 9x - 39 = 8x + 14 $.
## Step 3: Solve for $ x $
Subtract $ 8x $: $ x - 39 = 14 $. Add 39: $ x = 53 $. Then $ EF = 8(53) + 10 = 424 + 10 = 434 $. Still large. Maybe the diagram is $ D $---$ 4x $---$ E $---$ 8x + 10 $---$ F $, and $ DF = 9x - 39 $? Wait, original says "4?". Maybe a typo, but assuming $ DE = 4x $, then:
## Step 1: Segment addition
$ 9x - 39 = 4x + 8x + 10 $.
## Step 2: Simplify
$ 9x - 39 = 12x + 10 $.
## Step 3: Solve for $ x $
Subtract $ 9x $: $ -39 = 3x + 10 $. Subtract 10: $ -49 = 3x $. $ x = -\frac{49}{3} $, which is negative, impossible. So maybe the original is $ DE = 47 $ is wrong. Alternatively, maybe $ DF = 9x - 39 $, $ DE = 47 $, $ EF = 8x + 10 $, so $ 47 + 8x + 10 = 9x - 39 $ → $ 57 + 8x = 9x - 39 $ → $ 57 + 39 = x $ → $ x = 96 $, then $ EF = 8(96) + 10 = 778 $. But this is possible if lengths are large. So:
# Explanation (assuming $ DE = 47 $):
## Step 1: Segment addition
$ DF = DE + EF $ → $ 9x - 39 = 47 + 8x + 10 $.
## Step 2: Simplify
$ 9x - 39 = 8x + 57 $.
## Step 3: Solve for $ x $
$ x = 57 + 39 = 96 $.
## Step 4: Find $ EF $
$ EF = 8x + 10 = 8(96) + 10 = 778 $.
# Answer: $ 778 $ (assuming $ DE = 47 $; if $ DE $ is different, answer changes)

### Question 5: If $ UW = 6x - 35 $, find $ UW $. (Diagram: $ U $---$ 19 $---$ V $---$ 4x - 20 $---$ W $)
# Explanation:
## Step 1: Segment addition
$ UW = UV + VW $ → $ 6x - 35 = 19 + 4x - 20 $.
## Step 2: Simplify
$ 6x - 35 = 4x - 1 $.
## Step 3: Solve for $ x $
Subtract $ 4x $: $ 2x - 35 = -1 $. Add 35: $ 2x = 34 $. $ x = 17 $.
## Step 4: Find $ UW $
Substitute $ x = 17 $ into $ UW = 6x - 35 $: $ 6(17) - 35 = 102 - 35 = 67 $.
# Answer: $ 67 $

### Question 6: If $ HJ = 7x - 27 $, find the value of $ x $. (Diagram: $ H $---$ 3x - 5 $---$ I $---$ x - 1 $---$ J $)
# Explanation:
## Step 1: Segment addition
$ HJ = HI + IJ $ → $ 7x - 27 = (3x - 5) + (x - 1) $.
## Step 2: Simplify
$ 7x - 27 = 4x - 6 $.
## Step 3: Solve for $ x $
Subtract $ 4x $: $ 3x - 27 = -6 $. Add 27: $ 3x = 21 $. $ x = 7 $.
# Answer: $ 7 $

### Question 7: If $ BD = 7x - 10 $, $ BC = 4x - 29 $, and $ CD = 5x - 9 $, find each value. (Diagram: $ B $---$ C $---$ D $)
# Explanation:
## Step 1: Segment addition ( $ BD = BC + CD $)
$ 7x - 10 = (4x - 29) + (5x - 9) $.
## Step 2: Simplify
$ 7x - 10 = 9x - 38 $.
## Step 3: Solve for $ x $
Subtract $ 7x $: $ -10 = 2x - 38 $. Add 38: $ 28 = 2x $. $ x = 14 $.
## Step 4: Find $ BC $
Substitute $ x = 14 $ into $ BC = 4x - 29 $: $ 4(14) - 29 = 56 - 29 = 27 $.
## Step 5: Find $ CD $
Substitute $ x = 14 $ into $ CD = 5x - 9 $: $ 5(14) - 9 = 70 - 9 = 61 $.
## Step 6: Find $ BD $
Substitute $ x = 14 $ into $ BD = 7x - 10 $: $ 7(14) - 10 = 98 - 10 = 88 $.
# Answer:
$ x = 14 $
$ BC = 27 $
$ CD = 61 $
$ BD = 88 $

### Question 8: If $ \overline{BD} \cong \overline{BC} $, $ BD = 5x - 26 $, $ BC = 2x + 1 $, and $ AC = 43 $, find $ AB $. (Diagram: $ A $---$ B $---$ C $, $ D $ perpendicular to $ B $)
# Explanation:
## Step 1: Congruent segments mean $ BD = BC $
$ 5x - 26 = 2x + 1 $.
## Step 2: Solve for $ x $
Subtract $ 2x $: $ 3x - 26 = 1 $. Add 26: $ 3x = 27 $. $ x = 9 $.
## Step 3: Find $ BC $ (and $ BD $)
Substitute $ x = 9 $ into $ BC = 2x + 1 $: $ 2(9) + 1 = 19 $. So $ BC = 19 $.
## Step 4: Segment addition ( $ AC = AB + BC $)
$ 43 = AB + 19 $.
## Step 5: Solve for $ AB $
Subtract 19: $ AB = 43 - 19 = 24 $.
# Answer: $ 24 $

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QUESTION IMAGE

Question

Explanation:

Question 1: If \( LM = 22 \) and \( MN = 15 \), find \( LN \).

Step 1: Recall segment addition postulate

Step 2: Substitute values

Step 3: Calculate

Step 1: Recall segment addition postulate

Step 2: Substitute values

Step 3: Calculate

Step 1: Segment addition postulate

Step 2: Simplify equation

Step 3: Solve for \( x \)

Answer:

Question 2: If \( LN = 54 \) and \( LM = 31 \), find \( MN \).