find the value of each variable. simplify your answers as much as possible. if necessary, you may learn what…

Explanation:

Step1: Use similar - triangle property for the first pair

For the first pair of triangles, since the lines are parallel, the two triangles are similar. The ratios of corresponding sides are equal. We have the proportion $\frac{x}{9}=\frac{4}{8}$.

Step2: Solve for $x$

Cross - multiply the proportion $\frac{x}{9}=\frac{4}{8}$: $8x = 4\times9$, so $8x=36$, and $x=\frac{36}{8}=\frac{9}{2}$.

Step3: Use similar - triangle property for the second pair

For the second pair of triangles, since the angles are equal (marked as congruent), the two triangles are similar. The ratios of corresponding sides are equal. We have the proportion $\frac{y}{2}=\frac{6}{3}$.

Step4: Solve for $y$

Cross - multiply the proportion $\frac{y}{2}=\frac{6}{3}$: $3y = 6\times2$, so $3y = 12$, and $y = 4$.

Answer:

$x=\frac{9}{2}$, $y = 4$