In the diagram below, F is due east of E, and angle EFG=40°. Calculate the bearing of (a) F from G (b) E from G

The sum of angles in a triangle is $180^\circ$.

$$\angle EGF = 180^\circ - (\angle GEF + \angle EFG)$$ $$\angle EGF = 180^\circ - (60^\circ + 40^\circ)$$ $$\angle EGF = 180^\circ - 100^\circ$$ $$\angle EGF = 80^\circ$$

• Draw a North line at E.
• Since F is due east of E, the line EF points East. The angle from North at E, clockwise to EF is $090^\circ$.
• From the diagram, G is below the line EF. Angle GEF is $60^\circ$.
• Therefore, the line EG is $60^\circ$ clockwise from the line EF.
• Bearing of G from E = (Bearing of F from E) + $\angle GEF$

$$\text{Bearing (G from E)} = 090^\circ + 60^\circ = 150^\circ$$

• Draw a North line at F.
• Since E is due west of F, the line FE points West. The angle from North at F, clockwise to FE is $270^\circ$.
• From the diagram, G is below the line EF. Angle EFG is $40^\circ$.
• Therefore, the line FG is $40^\circ$ clockwise from the line FE.
• Bearing of G from F = (Bearing of E from F) + $\angle EFG$

$$\text{Bearing (G from F)} = 270^\circ + 40^\circ = 310^\circ$$

a) Calculate the bearing of F from G.
To find the bearing of F from G, we use the back bearing from G from F.
If the bearing of A from B is $\theta$:

• If $\theta < 180^\circ$, the bearing of B from A is $\theta + 180^\circ$.
• If $\theta > 180^\circ$, the bearing of B from A is $\theta - 180^\circ$.

We have Bearing (G from F) = $310^\circ$.
Since $310^\circ > 180^\circ$, the bearing of F from G is $310^\circ - 180^\circ$.

$$\text{Bearing (F from G)} = 310^\circ - 180^\circ = 130^\circ$$

This can also be found by drawing a North line at G and using parallel lines. The angle from North at F to FG is $310^\circ$. The angle from North at G to GF (alternate interior angle with the angle from the South line at F to FG) is $130^\circ$.

b) Calculate the bearing of E from G.
To find the bearing of E from G, we use the back bearing from G from E.
We have Bearing (G from E) = $150^\circ$.
Since $150^\circ < 180^\circ$, the bearing of E from G is $150^\circ + 180^\circ$.

$$\text{Bearing (E from G)} = 150^\circ + 180^\circ = 330^\circ$$

Final Answers:

a) The bearing of F from G is $\boxed{\text{130}^\circ}$

b) The bearing of E from G is $\boxed{\text{330}^\circ}$