find the tangent of ∠r. simplify your answer and write it as a proper fraction, improper fraction, or whole…

Explanation:

Step1: Identify right - triangle similarity

Since $\angle TQU=\angle R$ (corresponding angles as $TQ\parallel SR$), we can use the right - triangle $\triangle TQU$ to find $\tan(R)$. In right - triangle $\triangle TQU$, the opposite side to $\angle TQU$ is $TU$ and the adjacent side is $UQ$.

Step2: Recall the tangent formula

The tangent of an angle in a right - triangle is defined as $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. For $\angle TQU$ (which is equal to $\angle R$), the opposite side to $\angle TQU$ is $TU = 33$ and the adjacent side is $UQ=44$.

Step3: Calculate the tangent value

$\tan(R)=\tan(\angle TQU)=\frac{TU}{UQ}=\frac{33}{44}=\frac{3}{4}$

Answer:

$\frac{3}{4}$