Question

3.4.6 check your understanding
in the figure shown, $overline{hx}paralleloverline{sl},mangle wfl=(4(4x - 2))^{circ},mangle wkd=(4x - 12)^{circ},mangle fvw=(4y - 9)^{circ}$, and $mangle xdw=(6y - 41)^{circ}$.
find the values of $x$ and $y$.
$x = square$
$y=square$

Explanation:

Step1: Identify angle - relationship for x

Since $\overline{HX}\parallel\overline{SL}$, $\angle WFL$ and $\angle WKD$ are corresponding angles and are equal. So we set up the equation $4(4x - 2)=4x- 12$.
First, expand the left - hand side: $16x-8 = 4x - 12$.

Step2: Solve the equation for x

Subtract $4x$ from both sides: $16x-4x-8=4x - 4x-12$, which simplifies to $12x-8=-12$.
Then add 8 to both sides: $12x-8 + 8=-12 + 8$, giving $12x=-4$.
Divide both sides by 12: $x=-\frac{4}{12}=-\frac{1}{3}$.

Step3: Identify angle - relationship for y

$\angle FVW$ and $\angle XDW$ are vertical angles and are equal. So we set up the equation $4y-9 = 6y-41$.

Step4: Solve the equation for y

Subtract $4y$ from both sides: $4y-4y-9=6y-4y-41$, which simplifies to $-9 = 2y-41$.
Add 41 to both sides: $-9 + 41=2y-41 + 41$, giving $32 = 2y$.
Divide both sides by 2: $y = 16$.

Answer:

$x=-\frac{1}{3}$
$y = 16$