To solve the problem provided, let's break it down step-by-step:

Problem Summary:

In triangle $\triangle STU$:

• $s = 3.4 \, \text{cm}$
• $m\angle T = 165^\circ$
• $m\angle U = 13^\circ$

We are tasked to find the length of $t$, to the nearest tenth of a centimeter.

Solution Steps:

1. Calculate the Third Angle ($\angle S$): Using the triangle angle sum property: $m\angle S + m\angle T + m\angle U = 180^\circ$ Substituting values: $m\angle S = 180^\circ - 165^\circ - 13^\circ = 2^\circ$

2. Apply the Law of Sines: The Law of Sines states: $\frac{s}{\sin(S)} = \frac{t}{\sin(T)}$ Rearranging for $t$: $t = \frac{s \cdot \sin(T)}{\sin(S)}$

3. Substitute Known Values:

   • $s = 3.4 \, \text{cm}$
   • $\sin(T) = \sin(165^\circ)$
   • $\sin(S) = \sin(2^\circ)$

   Using a calculator:

   • $\sin(165^\circ) = \sin(180^\circ - 165^\circ) = \sin(15^\circ) \approx 0.2588$
   • $\sin(2^\circ) \approx 0.0349$

   Substitute into the formula: $t = \frac{3.4 \cdot 0.2588}{0.0349} \approx \frac{0.8800}{0.0349} \approx 25.2 \, \text{cm}$

Final Answer:

The length of $t$ is approximately 25.2 cm.

Would you like a detailed explanation of the Law of Sines or another example? Here are some related questions to explore:

1. What is the Law of Cosines, and how does it compare to the Law of Sines?
2. How do you calculate angles in a triangle using trigonometric functions?
3. How would the solution change if the triangle were not obtuse?
4. Why does the sum of angles in a triangle always equal $180^\circ$?
5. Can the Law of Sines be used in non-Euclidean geometry?

Tip: Always double-check the calculator's mode (degree or radian) when solving trigonometric problems!