geometry
1) find x if m∠kst = 160°, m∠rst = 37x - 5, and m∠rsk = 5x - 5.
2) m∠stg = x + 32, m∠gtu = x + 35, and m∠stu = 61°. find x.
3) find x if m∠deg = 17x - 3, m∠gef = 114°, and m∠def = 44x + 3.
solve for x.
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Question

Accepted Answer

1. **For problem 1**:
   - **Explanation**:
     - **Step 1: Use angle - addition property**
       - We know that $m\angle KST=m\angle RSK + m\angle RST$. Given $m\angle KST = 160^{\circ}$, $m\angle RST=37x - 5$, and $m\angle RSK = 5x - 5$. So, we can write the equation $160=(5x - 5)+(37x - 5)$.
     - **Step 2: Simplify the right - hand side of the equation**
       - Combine like terms: $(5x - 5)+(37x - 5)=5x+37x-5 - 5=42x-10$. So the equation becomes $160 = 42x-10$.
     - **Step 3: Solve for $x$**
       - Add 10 to both sides of the equation: $160 + 10=42x-10 + 10$, which gives $170 = 42x$. Then divide both sides by 42: $x=\frac{170}{42}=\frac{85}{21}\approx4.05$.
2. **For problem 2**:
   - **Explanation**:
     - **Step 1: Use angle - addition property**
       - We know that $m\angle STU=m\angle STG + m\angle GTU$. Given $m\angle STG=x + 32$, $m\angle GTU=x + 35$, and $m\angle STU = 61^{\circ}$. So, we can write the equation $61=(x + 32)+(x + 35)$.
     - **Step 2: Simplify the right - hand side of the equation**
       - Combine like terms: $(x + 32)+(x + 35)=x+x+32 + 35=2x+67$. So the equation becomes $61 = 2x+67$.
     - **Step 3: Solve for $x$**
       - Subtract 67 from both sides: $61-67=2x+67 - 67$, which gives $-6 = 2x$. Then divide both sides by 2: $x=-3$.
3. **For problem 3**:
   - **Explanation**:
     - **Step 1: Use angle - addition property**
       - We know that $m\angle DEF=m\angle DEG + m\angle GEF$. Given $m\angle DEG = 17x-3$, $m\angle GEF = 114^{\circ}$, and $m\angle DEF=44x + 3$. So, we can write the equation $44x + 3=(17x-3)+114$.
     - **Step 2: Simplify the right - hand side of the equation**
       - Combine like terms: $(17x-3)+114=17x+111$. So the equation becomes $44x + 3=17x+111$.
     - **Step 3: Solve for $x$**
       - Subtract $17x$ from both sides: $44x-17x + 3=17x-17x+111$, which gives $27x+3 = 111$. Then subtract 3 from both sides: $27x+3 - 3=111 - 3$, so $27x = 108$. Divide both sides by 27: $x = 4$.

We will solve the remaining problems in a similar fashion. For example, for problems involving line - segment lengths, if we have a line - segment $AB$ with a point $C$ on it such that $AB=AC + CB$, we can set up an equation based on the given expressions for $AB$, $AC$, and $CB$ and solve for $x$.

Let's take problem 4:
 - If we assume that the length of the whole line - segment is the sum of the lengths of its sub - segments. Let the whole line - segment be $TV$ and $TV=TS+SU+UV$. But since only $TS=x + 10$ and $SU = 2x+18$ are given (assuming $UV$ is not relevant for the equation setup based on the information available), we need more information to solve this problem completely.

For a general line - segment problem, say we have a line - segment with endpoints $M$ and $K$ and a point $L$ on it such that $MK=ML+LK$. If $MK = 17 + x$, $ML=2x + 16$, and $LK = 8$, then:
 - **Explanation**:
   - **Step 1: Set up the equation**
     - $17 + x=(2x + 16)+8$.
   - **Step 2: Simplify the right - hand side**
     - $(2x + 16)+8=2x+24$. So the equation is $17 + x=2x+24$.
   - **Step 3: Solve for $x$**
     - Subtract $x$ from both sides: $17 + x-x=2x-x+24$, which gives $17=x + 24$. Then subtract 24 from both sides: $17-24=x$, so $x=-7$.

We can continue this process for all the remaining line - segment and angle - related problems.

# Answer:
1. $x=\frac{85}{21}$
2. $x=-3$
3. $x = 4$
(Answers for remaining problems can be found by following the above - shown procedures for each individual problem)

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