11. find the value of x.
(6x + 7)° (8x - 17)°
12. find the value of x.
(11x - 15)° (5x - 13)°
13. if $overline{bd}perpoverline{ac}$, $mangle dbe=(2x - 1)°$, and $mangle cbe=(5x - 42)°$, find the value of x.
14. find the value of x if $overline{qs}$ bisects $angle pqr$ and $mangle pqr = 82°$.
(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)°
17. if $overline{np}$ bisects $angle mnq$, $mangle mnq=(8x + 12)°$, $mangle pnq = 78°$, and $mangle rnm=(3y - 9)°$, find the values of x and y.

Question

Accepted Answer

### 11.
# Explanation:
## Step1: Set up equation using vertical - angle property
Vertical angles are equal. So, $6x + 7=8x - 17$.
## Step2: Isolate $x$ 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 by 2: $x = 12$.
# Answer:
$x = 12$

### 12.
# Explanation:
## Step1: Set up equation using angle - addition property
The sum of the two angles is $11x-15=(5x - 13)$. This is incorrect. Assuming they are supplementary (if they form a straight - line), $(11x-15)+(5x - 13)=180$.
## Step2: Combine like terms
$11x+5x-15 - 13=180$, so $16x-28 = 180$.
## Step3: Isolate $x$ term
Add 28 to both sides: $16x=180 + 28=208$.
## Step4: Solve for $x$
Divide both sides by 16: $x=\frac{208}{16}=13$.
# Answer:
$x = 13$

### 13.
# Explanation:
## Step1: Use perpendicular - angle property
Since $\overline{BD}\perp\overline{AC}$, $\angle DBE+\angle CBE = 90^{\circ}$. So, $(2x - 1)+(5x - 42)=90$.
## Step2: Combine like terms
$2x+5x-1 - 42=90$, which gives $7x-43 = 90$.
## Step3: Isolate $x$ term
Add 43 to both sides: $7x=90 + 43=133$.
## Step4: Solve for $x$
Divide both sides by 7: $x = 19$.
# Answer:
$x = 19$

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

### 15.
# Explanation:
## Step1: Use vertical - angle property
The vertical angles are equal. So, $10x-61=x + 10$.
## Step2: Isolate $x$ terms
Subtract $x$ from both sides: $10x-x-61 = 10$, which gives $9x-61 = 10$.
## Step3: Solve for $x$
Add 61 to both sides: $9x=10 + 61=71$, so $x=\frac{71}{9}$.
For $y$, there is not enough information given in this part of the problem (assuming no other relationship is specified).
# Answer:
$x=\frac{71}{9}$, no value for $y$ with given info

### 16.
# Explanation:
## Step1: Use the fact that the sum of angles around a point is $360^{\circ}$ and there is a right - angle ($90^{\circ}$)
$(5x-17)+(3x - 11)+90+(2y + 5)=360$. First, combine like terms for $x$ terms: $5x+3x-17-11 + 90+2y + 5=360$, so $8x+2y+67 = 360$, or $8x+2y=293$. Also, if we assume the non - right angles are supplementary (if they are part of a linear pair), $(5x-17)+(3x - 11)=90$.
## Step2: Solve the equation for $x$ from the linear - pair
$5x+3x-17-11 = 90$, $8x-28 = 90$, add 28 to both sides: $8x=90 + 28 = 118$, $x=\frac{118}{8}=\frac{59}{4}$.
## Step3: Substitute $x$ into $8x + 2y=293$
$8\times\frac{59}{4}+2y=293$, $118+2y=293$, subtract 118 from both sides: $2y=293 - 118 = 175$, $y=\frac{175}{2}$.
# Answer:
$x=\frac{59}{4}$, $y=\frac{175}{2}$

### 17.
# Explanation:
## Step1: Use angle - bisector property for $x$
Since $\overline{NP}$ bisects $\angle MNQ$, $m\angle MNQ = 2m\angle PNQ$. So, $8x+12=2\times78$.
## Step2: Solve for $x$
$8x+12 = 156$, subtract 12 from both sides: $8x=156 - 12=144$, divide by 8: $x = 18$.
## Step3: Use vertical - angle or other angle - relationship for $y$ (assuming $\angle RNM$ and $\angle PNQ$ are vertical angles or supplementary in a known way)
If we assume they are vertical angles (if applicable), $3y-9 = 78$. Add 9 to both sides: $3y=78 + 9=87$, divide by 3: $y = 29$.
# Answer:
$x = 18$, $y = 29$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

11.

Step1: Set up equation using vertical - angle property

Step2: Isolate \(x\) terms

Step3: Solve for \(x\)

Step1: Set up equation using angle - addition property

Step2: Combine like terms

Step3: Isolate \(x\) term

Step4: Solve for \(x\)

Step1: Use perpendicular - angle property

Step2: Combine like terms

Step3: Isolate \(x\) term

Step4: Solve for \(x\)

Answer:

12.