Question

use the following information for questions 4 - 7. given (overline{xb}paralleloverline{tv}) where (overline{rw}) is a transversal. determine the value of the variables and each angle measure. 4. (mangle tkw=(8x - 20)^{circ}) and (mangle xhk=(3x + 2)^{circ}) 5. (mangle rhb=(2x - 20)^{circ}) and (mangle xhk = 44^{circ}) 6. (mangle hkt=(3y - 18)^{circ}), (mangle vkw=(8x + 12)^{circ}) and (mangle rhb=(2x + 18)^{circ}) 7. (mangle bhk=(8x + 12)^{circ}) and (mangle vkw = 140^{circ})

Explanation:

Step1: Identify angle - relationship for question 4

Since $\overline{XB}\parallel\overline{TV}$ and $\overline{RW}$ is a transversal, $\angle TKW$ and $\angle XHK$ are supplementary (same - side interior angles). So, $(8x - 20)+(3x + 2)=180$.
Combining like - terms: $8x+3x-20 + 2=180$, which simplifies to $11x-18 = 180$.
Adding 18 to both sides: $11x=180 + 18=198$.
Dividing both sides by 11: $x = 18$.
$m\angle TKW=8x-20=8\times18-20=144 - 20 = 124^{\circ}$.
$m\angle XHK=3x + 2=3\times18+2=54 + 2=56^{\circ}$.

Step2: Identify angle - relationship for question 5

$\angle RHB$ and $\angle XHK$ are vertical angles, so they are equal. Set $2x-20 = 44$.
Adding 20 to both sides: $2x=44 + 20=64$.
Dividing both sides by 2: $x = 32$.

Step3: Identify angle - relationship for question 6

$\angle HKT$ and $\angle VK W$ are vertical angles, so $\angle HKT=\angle VK W$. Also, $\angle RHB$ and $\angle VK W$ are corresponding angles. So, $2x + 18=8x+12$.
Subtracting $2x$ from both sides: $18=6x+12$.
Subtracting 12 from both sides: $6 = 6x$.
Dividing both sides by 6: $x = 1$.
Since $\angle HKT=\angle VK W$, and $\angle VK W=8x + 12=8\times1+12=20^{\circ}$, then $3y-18 = 20$.
Adding 18 to both sides: $3y=20 + 18=38$.
Dividing both sides by 3: $y=\frac{38}{3}$.

Step4: Identify angle - relationship for question 7

$\angle BHK$ and $\angle VK W$ are corresponding angles, so $8x+12 = 140$.
Subtracting 12 from both sides: $8x=140 - 12=128$.
Dividing both sides by 8: $x = 16$.

Answer:

1. $x = 18$, $m\angle TKW=124^{\circ}$, $m\angle XHK=56^{\circ}$
2. $x = 32$
3. $x = 1$, $y=\frac{38}{3}$
4. $x = 16$