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To solve this problem, we need to calculate \( \angle ABC \) based on the given bearings.

### Step 1: Interpret the bearings.
- **B is due south of A**, so the line \( AB \) is a vertical line pointing north-south.
- **C has a bearing of \( 050^\circ \) from B**, meaning the angle between the north direction at point B and the line \( BC \) is \( 50^\circ \) measured clockwise.
- **C has a bearing of \( 142^\circ \) from A**, meaning the angle between the north direction at point A and the line \( AC \) is \( 142^\circ \) measured clockwise.

### Step 2: Find \( \angle ABC \).
We can now calculate the angle \( \angle ABC \), which is the internal angle formed at point B between lines \( AB \) and \( BC \).

- The bearing of \( AC \) from A is \( 142^\circ \), so the angle between line \( AB \) (due south, which is \( 180^\circ \)) and line \( AC \) is:
  \[
  180^\circ - 142^\circ = 38^\circ.
  \]
- The bearing of \( BC \) from B is \( 50^\circ \), so the internal angle \( \angle ABC \) is:
  \[
  50^\circ - 38^\circ = 12^\circ.
  \]

### Final Answer:
\[
\boxed{12^\circ}.
\]

Would you like further details or have any questions about the calculation?

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Here are 5 related questions:
1. How are bearings measured, and why are they always given in three digits?
2. What is the significance of using clockwise directions in bearings?
3. How would the calculation change if point B was north of point A?
4. What trigonometric methods can be used to solve problems involving non-right triangles like this one?
5. How do you calculate angles in navigation when using a compass?

**Tip**: Bearings always start from the north and measure clockwise. Keep this in mind to avoid confusion in angles and directions!

B is due south of A. C has a bearing of 050° from B. C has a bearing of 142° from A. What is the value of ∠ABC?

This problem involves calculating the internal angle ∠ABC using bearings. Given bearings of 050° from B to C and 142° from A to C, the solution explores geometric principles and basic angle subtraction to find the final answer. Suitable for students in Grades 9-11.

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