THE BEARING of C from A is 148 degrees and the bearing of B  from C is 298 degrees and  B is directly south of A.
find angle ABC and draw a digram

To solve the problem, let's break it down into parts and then create a diagram.

### Step 1: Understanding the Bearings

- **Bearing of C from A**: This means that the angle from point A to point C is \( 148^\circ \) measured clockwise from the north.
- **Bearing of B from C**: This means that the angle from point C to point B is \( 298^\circ \) measured clockwise from the north.
- **B is directly south of A**: This means that the line AB is a vertical line, with B directly below A.

### Step 2: Analyzing the Situation

1. The bearing from A to C is \( 148^\circ \). This places C southeast of A.
2. The bearing from C to B is \( 298^\circ \). Since \( 298^\circ \) is equivalent to \( 298^\circ - 360^\circ = -62^\circ \), this suggests that B is to the northwest of C.
3. Since B is directly south of A, line AB is vertical.

### Step 3: Finding Angle ABC

To find the angle \( \angle ABC \):

1. Notice that the angle between north and AB is \( 180^\circ \) because B is directly south of A.
2. The bearing from C to B (or angle \( \angle BCA \)) can be calculated as the difference between \( 360^\circ \) and the bearing from C to B:

\[
\angle BCA = 360^\circ - 298^\circ = 62^\circ
\]

3. Since B is directly south of A, the line AB is vertical, and the bearing from A to B is \( 180^\circ \). The angle \( \angle BAC \) is the difference between the bearing of C from A and the bearing from A to B:

\[
\angle BAC = 180^\circ - 148^\circ = 32^\circ
\]

4. Finally, angle \( \angle ABC \) can be found as:

\[
\angle ABC = 180^\circ - \angle BAC - \angle BCA = 180^\circ - 32^\circ - 62^\circ = 86^\circ
\]

So, the angle \( \angle ABC \) is \( 86^\circ \).

### Diagram

I will describe how the diagram should look:

1. Draw a vertical line from A to B with B directly south of A.
2. Mark point C to the southeast of A such that the angle from the north line to AC is \( 148^\circ \).
3. Draw a line from C to B such that the angle from the north line at C to CB is \( 298^\circ \).

This will give you a triangle ABC where angle \( \angle ABC = 86^\circ \).

### Summary

- **Angle ABC**: \( 86^\circ \)
- **Diagram**: The diagram is a triangle ABC with B directly south of A, and C located southeast of A.

Would you like a visual representation of the diagram, or do you have any other questions?

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**Relative Questions:**
1. How do you calculate the bearing between two points if one is directly east of the other?
2. What is the bearing from B to C given that the bearing from C to B is 298°?
3. How does the concept of supplementary angles apply in solving problems related to bearings?
4. Can you find the length of side AC if AB is known and the angle ABC is 86°?
5. How would the scenario change if B were directly east of A instead of south?

**Tip:** Always remember that bearings are measured clockwise from the north and range from 0° to 360°.

Learn how to calculate angle ABC using bearings and direct positioning. Solve the problem step-by-step with detailed explanations and a diagram. Suitable for high school students studying geometry and trigonometry.

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