1) find ( mangle lmn ) if ( mangle lmf = 112^circ ) and ( mangle fmn = 54^circ ).
2) ( mangle ghi = 176^circ ) and ( mangle jhi = 140^circ ). find ( mangle ghj ).
3) ( mangle cba = 127^circ ) and ( mangle cbh = 55^circ ). find ( mangle hba ).
4) ( mangle efk = 30^circ ) and ( mangle efg = 138^circ ) find ( mangle kfg ).
5) ( mangle gfb = 42^circ ), ( mangle bfe = -2 + 11x ), and ( mangle gfe = 14x + 4 ). find ( x ).
6) ( mangle efu = x + 47 ), ( mangle efg = 162^circ ), and ( mangle ufg = x + 135 ). find ( x ).
7) ( mangle jhy = 6x + 10 ), ( mangle yhg = 69^circ ), and ( mangle jhg = 15x - 2 ). find ( mangle jhy ).
8) ( mangle qst = 72 + x ), ( mangle rst = 175^circ ), and ( mangle rsq = 117 + x ). find ( mangle rsq ).

Question

Accepted Answer

### Problem 1:
# Explanation:
## Step1: Identify angle relationship
We know that $ m\angle LMF = m\angle LMN + m\angle FMN $ (angle addition postulate). So we can solve for $ m\angle LMN $ by rearranging the formula: $ m\angle LMN = m\angle LMF - m\angle FMN $.
## Step2: Substitute values
Given $ m\angle LMF = 112^\circ $ and $ m\angle FMN = 54^\circ $, substitute into the formula: $ m\angle LMN = 112^\circ - 54^\circ $.
## Step3: Calculate the result
$ 112 - 54 = 58 $, so $ m\angle LMN = 58^\circ $.
# Answer:
$ 58^\circ $

### Problem 2:
# Explanation:
## Step1: Identify angle relationship
We know that $ m\angle GHI = m\angle GHJ + m\angle JHI $ (angle addition postulate). So we can solve for $ m\angle GHJ $ by rearranging the formula: $ m\angle GHJ = m\angle GHI - m\angle JHI $.
## Step2: Substitute values
Given $ m\angle GHI = 176^\circ $ and $ m\angle JHI = 140^\circ $, substitute into the formula: $ m\angle GHJ = 176^\circ - 140^\circ $.
## Step3: Calculate the result
$ 176 - 140 = 36 $, so $ m\angle GHJ = 36^\circ $.
# Answer:
$ 36^\circ $

### Problem 3:
# Explanation:
## Step1: Identify angle relationship
We know that $ m\angle CBA = m\angle CBH + m\angle HBA $ (angle addition postulate). So we can solve for $ m\angle HBA $ by rearranging the formula: $ m\angle HBA = m\angle CBA - m\angle CBH $.
## Step2: Substitute values
Given $ m\angle CBA = 127^\circ $ and $ m\angle CBH = 55^\circ $, substitute into the formula: $ m\angle HBA = 127^\circ - 55^\circ $.
## Step3: Calculate the result
$ 127 - 55 = 72 $, so $ m\angle HBA = 72^\circ $.
# Answer:
$ 72^\circ $

### Problem 4:
# Explanation:
## Step1: Identify angle relationship
We know that $ m\angle EFG = m\angle EFK + m\angle KFG $ (angle addition postulate). So we can solve for $ m\angle KFG $ by rearranging the formula: $ m\angle KFG = m\angle EFG - m\angle EFK $.
## Step2: Substitute values
Given $ m\angle EFK = 30^\circ $ and $ m\angle EFG = 138^\circ $, substitute into the formula: $ m\angle KFG = 138^\circ - 30^\circ $.
## Step3: Calculate the result
$ 138 - 30 = 108 $, so $ m\angle KFG = 108^\circ $.
# Answer:
$ 108^\circ $

### Problem 5:
# Explanation:
## Step1: Identify angle relationship
We know that $ m\angle GFE = m\angle GFB + m\angle BFE $ (angle addition postulate). So we can set up the equation: $ 14x + 4 = 42 + (-2 + 11x) $.
## Step2: Simplify the equation
Simplify the right - hand side: $ 14x + 4 = 42 - 2+11x $, which becomes $ 14x + 4 = 40 + 11x $.
## Step3: Solve for x
Subtract $ 11x $ from both sides: $ 14x - 11x+4 = 40 + 11x - 11x $, so $ 3x+4 = 40 $. Then subtract 4 from both sides: $ 3x+4 - 4 = 40 - 4 $, which gives $ 3x = 36 $. Divide both sides by 3: $ x=\frac{36}{3}=12 $.
# Answer:
$ x = 12 $

### Problem 6:
# Explanation:
## Step1: Identify angle relationship
We know that $ m\angle EFG = m\angle EFU + m\angle UFG $ (angle addition postulate). So we can set up the equation: $ 162=(x + 47)+(x + 135) $.
## Step2: Simplify the equation
Simplify the right - hand side: $ 162=x + 47+x + 135 $, which becomes $ 162 = 2x+182 $.
## Step3: Solve for x
Subtract 182 from both sides: $ 162-182=2x + 182-182 $, so $ - 20 = 2x $. Divide both sides by 2: $ x=\frac{-20}{2}=-10 $.
# Answer:
$ x=-10 $

### Problem 7:
# Explanation:
## Step1: Identify angle relationship
We know that $ m\angle IHG = m\angle IHY + m\angle YHG $ (angle addition postulate). So we can set up the equation: $ 15x - 2=(6x + 10)+69 $.
## Step2: Simplify the equation
Simplify the right - hand side: $ 15x - 2=6x+10 + 69 $, which becomes $ 15x - 2=6x + 79 $.
## Step3: Solve for x
Subtract $ 6x $ from both sides: $ 15x-6x - 2=6x - 6x+79 $, so $ 9x - 2 = 79 $. Add 2 to both sides: $ 9x-2 + 2=79 + 2 $, which gives $ 9x = 81 $. Divide both sides by 9: $ x=\frac{81}{9}=9 $.
## Step4: Find $ m\angle IHY $
Substitute $ x = 9 $ into $ m\angle IHY=6x + 10 $: $ m\angle IHY=6\times9 + 10=54 + 10 = 64^\circ $.
# Answer:
$ 64^\circ $

### Problem 8:
# Explanation:
## Step1: Identify angle relationship
We know that $ m\angle RST = m\angle RSQ + m\angle QST $ (angle addition postulate). So we can set up the equation: $ 175=(117 + x)+(72 + x) $.
## Step2: Simplify the equation
Simplify the right - hand side: $ 175=117 + x+72 + x $, which becomes $ 175 = 2x+189 $.
## Step3: Solve for x
Subtract 189 from both sides: $ 175 - 189=2x+189 - 189 $, so $ - 14 = 2x $. Divide both sides by 2: $ x=\frac{-14}{2}=-7 $.
## Step4: Find $ m\angle RSQ $
Substitute $ x=-7 $ into $ m\angle RSQ = 117 + x $: $ m\angle RSQ=117+(-7)=110^\circ $.
# Answer:
$ 110^\circ $

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QUESTION IMAGE

Question

Explanation:

Problem 1:

Step1: Identify angle relationship

Step2: Substitute values

Step3: Calculate the result

Step1: Identify angle relationship

Step2: Substitute values

Step3: Calculate the result

Step1: Identify angle relationship

Step2: Substitute values

Step3: Calculate the result

Answer:

Problem 2: