find the tangent of $angle u$.\nsimplify your answer and write it as a proper fraction, improper fraction…

Explanation:

Step1: Identify similar triangles

The two right triangles are similar: $\triangle PQR \sim \triangle TSU$, so corresponding angles are equal ($\angle P = \angle U$), so $\tan(U) = \tan(P)$.

Step2: Recall tangent definition

For a right triangle, $\tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}}$.

Step3: Substitute sides for $\angle P$

For $\angle P$, opposite side = $QR = 65$, adjacent side = $PQ = 72$. $\tan(P) = \frac{65}{72}$

Step4: Equate to $\tan(U)$

Since $\angle U = \angle P$, $\tan(U) = \tan(P)$

Answer:

$\frac{65}{72}$