List the angle measures of △UVW in order from smallest to largest. Assume that x is a positive number.59x48x56xVUW

To find the angle measures of triangle $\triangle UVW$ in order from smallest to largest, we will follow these steps:

1. ### Break Down the Problem

We have three angles given in terms of $x$:

• Angle $U = 59x$
• Angle $V = 48x$
• Angle $W = 56x$

2. ### Relevant Concepts

In any triangle, the sum of the angles is equal to $180^\circ$. Therefore, we can set up the equation: $59x + 48x + 56x = 180$

3. ### Analysis and Detail

1. Combine the angles: $59x + 48x + 56x = 163x$

2. Set the equation equal to $180^\circ$: $163x = 180$

3. Solve for $x$: $x = \frac{180}{163} \approx 1.104$

4. Calculate each angle:

   • $U = 59x = 59 \times \frac{180}{163} \approx 65.03^\circ$
   • $V = 48x = 48 \times \frac{180}{163} \approx 52.91^\circ$
   • $W = 56x = 56 \times \frac{180}{163} \approx 61.65^\circ$

4. ### Verify and Summarize

Confirm the angles by checking their sum: $65.03 + 52.91 + 61.65 \approx 179.59 \approx 180^\circ$ This confirms that the calculations are accurate.

Final Answer

The angle measures of triangle $\triangle UVW$ in order from smallest to largest are:

1. $V \approx 52.91^\circ$
2. $W \approx 61.65^\circ$
3. $U \approx 65.03^\circ$