Question

in the rectangle below, $su = 6x - 2$, $rv = 17$, and $m\angle svt = 62^\circ$.
find the value of $x$ and $m\angle vur$.

Explanation:

Step1: Use rectangle diagonal property

In a rectangle, diagonals are equal and bisect each other, so $SU = 2 \times RV$.
Substitute values: $6x - 2 = 2 \times 17$

Step2: Solve for x

Calculate right-hand side: $6x - 2 = 34$
Add 2 to both sides: $6x = 36$
Divide by 6: $x = \frac{36}{6} = 6$

Step3: Find $\angle VUR$

First, $\angle SVT$ and $\angle UV R$ are vertical angles, so $m\angle UV R = 62^\circ$.
Since $RV = UV$, $\triangle UV R$ is isosceles, so $\angle VUR = \angle VRU$.
Sum of angles in a triangle: $m\angle VUR + m\angle VRU + m\angle UV R = 180^\circ$
Let $m\angle VUR = y$, then $2y + 62^\circ = 180^\circ$

Step4: Calculate angle measure

Subtract $62^\circ$: $2y = 180^\circ - 62^\circ = 118^\circ$
Divide by 2: $y = \frac{118^\circ}{2} = 59^\circ$

Answer:

$x = 6$
$m\angle VUR = 59^\circ$