date: ______ per: ______ unit 1: geometry basics
homework 2: segment addition postulate
** this is a 2 - page document **

use the diagram below to answer questions 1 and 2.
1. if $ lm = 22 $ and $ mn = 15 $, find $ ln $.
(diagram: ( l---m---n ))
2. if $ ln = 54 $ and $ lm = 31 $, find $ mn $.

3. if $ rt = 36 $, find the value of $ x $.
(diagram: ( r---s---t ), ( rs = 6x + 1 ), ( st = x + 7 ); handwritten: ( 6x + 1 = x + 7 ))
4. if $ df = 9x - 39 $, find $ ef $.
(diagram: ( d---e---f ), ( de = 47 ), ( ef = 3x + 10 ))

5. if $ uw = 6x - 35 $, find $ uw $.
(diagram: ( u---v---w ), ( uv = 19 ), ( vw = 4x - 20 ))
6. if $ hj = 7x - 27 $, find the value of $ x $.
(diagram: ( h---i---j ), ( hi = 3x - 5 ), ( ij = x - 1 ))

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 ), ( d ) above ( b ) (right angle))

Question

date: ______ per: ______ unit 1: geometry basics
homework 2: segment addition postulate
** this is a 2 - page document **

use the diagram below to answer questions 1 and 2.
1. if $ lm = 22 $ and $ mn = 15 $, find $ ln $.
   (diagram: ( l---m---n ))
2. if $ ln = 54 $ and $ lm = 31 $, find $ mn $.

3. if $ rt = 36 $, find the value of $ x $.
   (diagram: ( r---s---t ), ( rs = 6x + 1 ), ( st = x + 7 ); handwritten: ( 6x + 1 = x + 7 ))
4. if $ df = 9x - 39 $, find $ ef $.
   (diagram: ( d---e---f ), ( de = 47 ), ( ef = 3x + 10 ))

5. if $ uw = 6x - 35 $, find $ uw $.
   (diagram: ( u---v---w ), ( uv = 19 ), ( vw = 4x - 20 ))
6. if $ hj = 7x - 27 $, find the value of $ x $.
   (diagram: ( h---i---j ), ( hi = 3x - 5 ), ( ij = x - 1 ))

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 ), ( d ) above ( b ) (right angle))

Accepted Answer

### Question 1: If $ LM = 22 $ and $ MN = 15 $, find $ LN $.
# Explanation:
## Step1: Apply Segment Addition Postulate
By the Segment Addition Postulate, $ LN = LM + MN $.
## Step2: Substitute the given values
Substitute $ LM = 22 $ and $ MN = 15 $ into the formula: $ LN = 22 + 15 $.
## Step3: Calculate the sum
$ 22 + 15 = 37 $.
# Answer:
$ LN = 37 $

### Question 2: If $ LN = 54 $ and $ LM = 31 $, find $ MN $.
# Explanation:
## Step1: Apply Segment Addition Postulate
By the Segment Addition Postulate, $ LN = LM + MN $, so we can solve for $ MN $ as $ MN = LN - LM $.
## Step2: Substitute the given values
Substitute $ LN = 54 $ and $ LM = 31 $ into the formula: $ MN = 54 - 31 $.
## Step3: Calculate the difference
$ 54 - 31 = 23 $.
# Answer:
$ MN = 23 $

### Question 3: If $ RT = 36 $, find the value of $ x $.
# Explanation:
## Step1: Apply Segment Addition Postulate
By the Segment Addition Postulate, $ RT = RB + BT $. From the diagram, $ RB = 6x + 1 $ and $ BT = x + 7 $, so $ 6x + 1 + x + 7 = 36 $.
## Step2: Combine like terms
Combine the $ x $-terms and constant terms: $ 7x + 8 = 36 $.
## Step3: Solve for $ x $
Subtract 8 from both sides: $ 7x = 36 - 8 = 28 $. Then divide both sides by 7: $ x = \frac{28}{7} = 4 $.
# Answer:
$ x = 4 $

### Question 4: If $ DF = 9x - 39 $, find $ EF $.
# Explanation:
## Step1: Apply Segment Addition Postulate
By the Segment Addition Postulate, $ DF = DE + EF $. From the diagram, $ DE = 47 $ and $ EF = 3x + 10 $, so $ 47 + 3x + 10 = 9x - 39 $.
## Step2: Combine like terms
Combine the constant terms on the left: $ 57 + 3x = 9x - 39 $.
## Step3: Solve for $ x $
Subtract $ 3x $ from both sides: $ 57 = 6x - 39 $. Add 39 to both sides: $ 6x = 57 + 39 = 96 $. Divide both sides by 6: $ x = \frac{96}{6} = 16 $.
## Step4: Find $ EF $
Substitute $ x = 16 $ into $ EF = 3x + 10 $: $ EF = 3(16) + 10 = 48 + 10 = 58 $.
# Answer:
$ EF = 58 $

### Question 5: If $ UW = 6x - 35 $, find $ UW $.
# Explanation:
## Step1: Apply Segment Addition Postulate
By the Segment Addition Postulate, $ UW = UV + VW $. From the diagram, $ UV = 19 $ and $ VW = 4x - 20 $, so $ 19 + 4x - 20 = 6x - 35 $.
## Step2: Simplify the left side
Simplify $ 19 - 20 + 4x = 4x - 1 $, so the equation becomes $ 4x - 1 = 6x - 35 $.
## Step3: Solve for $ x $
Subtract $ 4x $ from both sides: $ -1 = 2x - 35 $. Add 35 to both sides: $ 2x = 34 $. Divide both sides by 2: $ x = 17 $.
## Step4: Find $ UW $
Substitute $ x = 17 $ into $ UW = 6x - 35 $: $ UW = 6(17) - 35 = 102 - 35 = 67 $.
# Answer:
$ UW = 67 $

### Question 6: If $ HJ = 7x - 27 $, find the value of $ x $.
# Explanation:
## Step1: Apply Segment Addition Postulate
By the Segment Addition Postulate, $ HJ = HI + IJ $. From the diagram, $ HI = 3x - 5 $ and $ IJ = x - 1 $, so $ 3x - 5 + x - 1 = 7x - 27 $.
## Step2: Combine like terms
Combine the $ x $-terms and constant terms on the left: $ 4x - 6 = 7x - 27 $.
## Step3: Solve for $ x $
Subtract $ 4x $ from both sides: $ -6 = 3x - 27 $. Add 27 to both sides: $ 3x = 21 $. Divide both sides by 3: $ x = 7 $.
# Answer:
$ x = 7 $

### Question 7: If $ BD = 7x - 10 $, $ BC = 4x - 29 $, and $ CD = 5x - 9 $, find each value.
# Explanation:
## Step1: Apply Segment Addition Postulate
By the Segment Addition Postulate, $ BD = BC + CD $. So $ 7x - 10 = (4x - 29) + (5x - 9) $.
## Step2: Simplify the right side
Combine like terms on the right: $ 4x + 5x - 29 - 9 = 9x - 38 $, so the equation becomes $ 7x - 10 = 9x - 38 $.
## Step3: Solve for $ x $
Subtract $ 7x $ from both sides: $ -10 = 2x - 38 $. Add 38 to both sides: $ 2x = 28 $. Divide both sides by 2: $ x = 14 $.
## Step4: Find $ BC $
Substitute $ x = 14 $ into $ BC = 4x - 29 $: $ BC = 4(14) - 29 = 56 - 29 = 27 $.
## Step5: Find $ CD $
Substitute $ x = 14 $ into $ CD = 5x - 9 $: $ CD = 5(14) - 9 = 70 - 9 = 61 $.
## Step6: Find $ BD $
Substitute $ x = 14 $ into $ BD = 7x - 10 $: $ BD = 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 $.
# Explanation:
## Step1: Use the congruence of segments
Since $ \overline{BD} \cong \overline{BC} $, their lengths are equal, so $ 5x - 26 = 2x + 1 $.
## Step2: Solve for $ x $
Subtract $ 2x $ from both sides: $ 3x - 26 = 1 $. Add 26 to both sides: $ 3x = 27 $. Divide both sides by 3: $ x = 9 $.
## Step3: Find $ BC $ (and $ BD $)
Substitute $ x = 9 $ into $ BC = 2x + 1 $: $ BC = 2(9) + 1 = 19 $. So $ BD = 19 $ as well.
## Step4: Apply Segment Addition Postulate for $ AC $
By the Segment Addition Postulate, $ AC = AB + BC $. We know $ AC = 43 $ and $ BC = 19 $, so $ 43 = AB + 19 $.
## Step5: Solve for $ AB $
Subtract 19 from both sides: $ AB = 43 - 19 = 24 $.
# Answer:
$ AB = 24 $

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QUESTION IMAGE

Question

Explanation:

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

Step1: Apply Segment Addition Postulate

Step2: Substitute the given values

Step3: Calculate the sum

Step1: Apply Segment Addition Postulate

Step2: Substitute the given values

Step3: Calculate the difference

Step1: Apply Segment Addition Postulate

Step2: Combine like terms

Step3: Solve for \( x \)

Answer:

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