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To determine the most probable values for the three interior angles of a triangle, given their measured values and uncertainties, we need to follow these steps:

1. **Calculate the sum of measured angles:** 
   The sum of interior angles in a triangle should be exactly \(180^\circ\). If the measured angles do not sum to \(180^\circ\), we need to adjust them proportionally.

2. **Convert angles to decimal form:** 
   First, we convert each angle to decimal degrees for easier calculations. Here are the given measurements:
   - Angle 1: \(58^\circ 26' 48''\)
   - Angle 2: \(67^\circ 06' 56''\)
   - Angle 3: \(54^\circ 26' 24''\)

   Converting each angle:
   \[
   \text{Angle 1} = 58^\circ + \frac{26'}{60} + \frac{48''}{3600} = 58.4467^\circ
   \]
   \[
   \text{Angle 2} = 67^\circ + \frac{6'}{60} + \frac{56''}{3600} = 67.1156^\circ
   \]
   \[
   \text{Angle 3} = 54^\circ + \frac{26'}{60} + \frac{24''}{3600} = 54.4400^\circ
   \]

3. **Calculate the sum of measured angles:**
   Adding these angles:
   \[
   58.4467^\circ + 67.1156^\circ + 54.4400^\circ = 179.9943^\circ
   \]

   Since this sum is close but not exactly \(180^\circ\), we’ll need to make a small adjustment.

4. **Adjust the angles proportionally:**
   To achieve a total of \(180^\circ\), we need an increase of \(0.0057^\circ\) (i.e., \(180 - 179.9943 = 0.0057^\circ\)).

   Distribute this adjustment proportionally across the three angles based on their standard errors:
   \[
   \text{Adjusted Angle 1} = 58.4467^\circ + \frac{5.1}{5.1 + 4.3 + 2.6} \times 0.0057
   \]
   \[
   \text{Adjusted Angle 2} = 67.1156^\circ + \frac{4.3}{5.1 + 4.3 + 2.6} \times 0.0057
   \]
   \[
   \text{Adjusted Angle 3} = 54.4400^\circ + \frac{2.6}{5.1 + 4.3 + 2.6} \times 0.0057
   \]

5. **Convert back to degrees, minutes, and seconds.**

Would you like a detailed calculation of each adjusted angle or have any questions?

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**Related Questions:**

1. How do you handle uncertainties when measuring angles in other geometric figures?
2. What are other common methods for adjusting measurement data with uncertainties?
3. How do different types of errors (systematic vs. random) affect angle measurements?
4. How does the process change if the angle measurements sum to more than 180°?
5. What other applications exist for proportional adjustments in error analysis?

**Tip:** When converting angles from degrees, minutes, and seconds to decimal form, each minute is 1/60 of a degree, and each second is 1/3600 of a degree.

Determine the most probable values for the three interior angles of a triangle that were measured as follows: Angle 1 = 58°26'48" with ±5.1" uncertainty, Angle 2 = 67°06'56" with ±4.3" uncertainty, and Angle 3 = 54°26'24" with ±2.6" uncertainty.

Learn how to determine the most probable values for three interior angles of a triangle given their measurements and uncertainties. This solution involves error analysis, angle measurement adjustment, and the proportional adjustment formula, suitable for students in Grades 10-12.

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