Question

1. what is xy?
2. what is m∠r?

Explanation:

Step1: Identify similar - triangles

Triangles $UVW$ and $XYZ$ are similar because they have two equal angles ($23^{\circ}$ and $90^{\circ}$). The ratio of corresponding sides of similar triangles is equal.

Step2: Set up the proportion

The ratio of the side opposite the $23^{\circ}$ angle to the side opposite the $90^{\circ}$ angle in $\triangle UVW$ is the same as in $\triangle XYZ$. In $\triangle UVW$, the side opposite the $23^{\circ}$ angle is $24$ ft and the side opposite the $90^{\circ}$ angle is $26$ ft. In $\triangle XYZ$, the side opposite the $90^{\circ}$ angle is $XY$ and the side opposite the $23^{\circ}$ angle is $10$ ft. So, $\frac{24}{26}=\frac{10}{XY}$.

Step3: Solve the proportion for $XY$

Cross - multiply: $24\times XY = 26\times10$. Then $XY=\frac{26\times10}{24}=\frac{260}{24}=\frac{65}{6}\approx10.83$ ft.

For the second question about $\angle R$:

Step1: Use angle - sum property of polygons

The two polygons are congruent (since their corresponding sides and angles are equal as indicated by the markings). In polygon $DEFG$, we can find the missing angle. The sum of interior angles of a quadrilateral is $(4 - 2)\times180^{\circ}=360^{\circ}$.

Step2: Calculate the missing angle in $DEFG$

Let the missing angle in $DEFG$ be $\theta$. Then $\theta=360-(80 + 107+62)=360 - 249 = 111^{\circ}$. Since the polygons are congruent, $\angle R$ in polygon $RSTU$ corresponds to the $111^{\circ}$ angle in $DEFG$. So $m\angle R = 111^{\circ}$.

Answer:

$XY=\frac{65}{6}\text{ ft}\approx10.83\text{ ft}$
$m\angle R = 111^{\circ}$