Question

3.4.6 check your understanding
in the figure shown, $overline{hx}paralleloverline{sl}$, $mangle lfw=(4(2x + 5))^{circ}$, $mangle dkw=(3x + 17)^{circ}$, $mangle svw=(6y + 28)^{circ}$, and $mangle kdw=(2y-16)^{circ}$.
determine the values of $x$ and $y$.
$x = square$
$y=square$

Explanation:

Step1: Identify equal - angle relationship

Since $\overline{HX}\parallel\overline{SL}$, $\angle LFW$ and $\angle DKW$ are corresponding angles, so $\angle LFW=\angle DKW$.
$4(2x + 5)=3x+17$

Step2: Expand the left - hand side

Expand $4(2x + 5)$ using the distributive property: $8x+20 = 3x + 17$.

Step3: Solve for $x$

Subtract $3x$ from both sides: $8x-3x+20=3x - 3x+17$, which gives $5x+20 = 17$.
Then subtract 20 from both sides: $5x+20 - 20=17 - 20$, so $5x=-3$.
Divide both sides by 5: $x=-\frac{3}{5}$.

Step4: Identify another equal - angle relationship

$\angle SVW$ and $\angle KDW$ are same - side interior angles, so $\angle SVW+\angle KDW = 180^{\circ}$.
$(6y + 28)+(2y-16)=180$.

Step5: Combine like terms

Combine the $y$ terms and the constant terms: $6y+2y+28 - 16=180$, which gives $8y + 12=180$.

Step6: Solve for $y$

Subtract 12 from both sides: $8y+12 - 12=180 - 12$, so $8y=168$.
Divide both sides by 8: $y = 21$.

Answer:

$x=-\frac{3}{5}$
$y = 21$