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To determine the angle \( \angle BYO \) in the given diagram, we start by understanding the directional bearings provided:

1. \( O \) bears 164° from \( Y \): This indicates that the direction from \( Y \) to \( O \) is 164° clockwise from the north.
2. \( B \) bears 081° from \( Y \): This suggests that the direction from \( Y \) to \( B \) is 081° clockwise from the north.
3. \( B \) bears 007° from \( O \): The direction from \( O \) to \( B \) is 007° clockwise from the north.

To find \( \angle BYO \), we need to calculate the difference in the bearings from \( Y \) to \( O \) and from \( Y \) to \( B \):

- \( \text{Bearing from Y to O} = 164^\circ \)
- \( \text{Bearing from Y to B} = 081^\circ \)

Since the bearing from \( Y \) to \( O \) is greater than from \( Y \) to \( B \), \( \angle BYO \) is given by the subtraction of the two angles:

\[
\angle BYO = 164^\circ - 081^\circ = 83^\circ
\]

Therefore, \( \angle BYO = 83^\circ \).

Do you have any questions on this solution or need more details on how the angles were calculated?

1. How does the bearing system work?
2. Can you explain why the higher bearing number was subtracted from the lower one?
3. What if the angles were reversed, how would the calculation change?
4. How can bearings affect the layout of a triangle in navigation?
5. Could you solve for the other angles of the triangle using similar methods?

**Tip:** When dealing with bearings in navigation problems, always visualize the bearings as angles measured clockwise from north to ensure clarity in the directionality of the calculations.

O bears 164° from Y. B bears 081° and 007° from Y and O respectively. Find the angle BYO.

Learn how to calculate the angle BYO given that O bears 164° from Y and B bears 081° and 007° from Y and O respectively. This problem covers bearings and angle calculation, ideal for students in Grades 9-12.

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