11. find the value of x.
(6x + 7)° (8x - 17)°
12. find the value of x.
(11x - 15)° (5x - 13)°
13. if bd ⊥ ac, m∠dbe=(2x - 1)°, and m∠cbe=(5x - 42)°, find the value of x.
14. find the value of x if qs bisects ∠pqr and m∠pqr = 82°.
(10x + 1)°
15. find the values of x and y.
(18y + 5)° (10x - 61)° (x + 10)°
16. find the values of x and y.
(2y + 5)° (5x - 17)° (3x - 11)°

Question

Accepted Answer

### 11.
# Explanation:
## Step1: Set the angles equal
Since vertical angles are equal, we set $6x + 7=8x - 17$.
## Step2: Isolate the variable terms
Subtract $6x$ from both sides: $7 = 8x-6x - 17$, which simplifies to $7=2x - 17$.
## Step3: Solve for $x$
Add 17 to both sides: $7 + 17=2x$, so $24 = 2x$. Then divide both sides by 2, $x = 12$.
# Answer:
$x = 12$

### 12.
# Explanation:
## Step1: Set up the equation
Since the two angles are supplementary (linear - pair), $(11x - 15)+(5x - 13)=180$.
## Step2: Combine like - terms
$11x+5x-15 - 13 = 180$, which gives $16x-28 = 180$.
## Step3: Isolate the variable term
Add 28 to both sides: $16x=180 + 28$, so $16x = 208$.
## Step4: Solve for $x$
Divide both sides by 16: $x=\frac{208}{16}=13$.
# Answer:
$x = 13$

### 13.
# Explanation:
## Step1: Set up the equation
Since $\overline{BD}\perp\overline{AC}$, $\angle DBE$ and $\angle CBE$ are complementary, so $(2x - 1)+(5x - 42)=90$.
## Step2: Combine like - terms
$2x+5x-1 - 42 = 90$, which gives $7x-43 = 90$.
## Step3: Isolate the variable term
Add 43 to both sides: $7x=90 + 43$, so $7x = 133$.
## Step4: Solve for $x$
Divide both sides by 7: $x = 19$.
# Answer:
$x = 19$

### 14.
# Explanation:
## Step1: Use the angle - bisector property
Since $\overline{QS}$ bisects $\angle PQR$, $\angle PQS=\frac{1}{2}\angle PQR$. So $10x + 1=\frac{82}{2}$.
## Step2: Simplify the right - hand side
$10x + 1 = 41$.
## Step3: Solve for $x$
Subtract 1 from both sides: $10x=41 - 1$, so $10x = 40$. Then divide both sides by 10, $x = 4$.
# Answer:
$x = 4$

### 15.
# Explanation:
## Step1: Set up the equation for vertical angles
$(10x - 61)=(x + 10)$ (vertical angles are equal).
## Step2: Solve for $x$
Subtract $x$ from both sides: $10x-x-61 = 10$, so $9x-61 = 10$. Add 61 to both sides: $9x=10 + 61$, so $9x = 71$, and $x=\frac{71}{9}$.
## Step3: Since the lines are perpendicular (assumed from the angle relationship), we can't find $y$ from the given information about $x$ angles only. But if we assume the other pair of angles are also related in a way that we can find $y$. Let's assume the angles are part of a right - angled intersection. Then $(18y + 5)+(x + 10)=90$. Substitute $x=\frac{71}{9}$ into the equation: $18y+5+\frac{71}{9}+10 = 90$. First, combine the constant terms: $18y+\frac{45 + 71+90}{9}=90$. $18y+\frac{206}{9}=90$. Subtract $\frac{206}{9}$ from both sides: $18y=90-\frac{206}{9}=\frac{810 - 206}{9}=\frac{604}{9}$. Then $y=\frac{604}{9	imes18}=\frac{302}{81}$.
# Answer:
$x=\frac{71}{9},y = \frac{302}{81}$

### 16.
# Explanation:
## Step1: Set up the equation for vertical angles
$(5x - 17)=(3x - 11)$ (vertical angles are equal).
## Step2: Solve for $x$
Subtract $3x$ from both sides: $5x-3x-17=-11$, so $2x-17=-11$. Add 17 to both sides: $2x=-11 + 17$, so $2x = 6$, and $x = 3$.
## Step3: Since the lines are perpendicular (right - angle symbol), $(2y + 5)+(5x - 17)=90$. Substitute $x = 3$ into the equation: $(2y + 5)+(5	imes3-17)=90$. First, simplify the second term: $(2y + 5)+(15 - 17)=90$, so $(2y + 5)-2 = 90$. Then $2y+3 = 90$. Subtract 3 from both sides: $2y=90 - 3$, so $2y = 87$, and $y=\frac{87}{2}$.
# Answer:
$x = 3,y=\frac{87}{2}$

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QUESTION IMAGE

Question

Explanation:

11.

Step1: Set the angles equal

Step2: Isolate the variable terms

Step3: Solve for $x$

Step1: Set up the equation

Step2: Combine like - terms

Step3: Isolate the variable term

Step4: Solve for $x$

Step1: Set up the equation

Step2: Combine like - terms

Step3: Isolate the variable term

Step4: Solve for $x$

Answer:

12.