Suspension bridge triangle ABC; span AB

Method. Mean directions are obtained from the two faces, angles from differences of directions, the triangle sum is checked against 180°, the error is distributed equally, and the side AB is then found by the sine rule.

Check and correction

Sum of angles = 179°58′35″; required = 180°00′00″; error = -0°01′25″ (sum is less than 180°). The permissible error is about the least count times $\sqrt{3}$ = 1′ × 1.73 ≈ 1′44″, so the error is acceptable and is distributed equally: correction = +0°00′28.33″ per angle.

Span AB

In triangle ABC, the side AB is opposite to the angle C and BC is opposite to angle A.

$$\begin{aligned} \frac{AB}{\sin C} &= \frac{BC}{\sin A}\\ AB &= BC\,\frac{\sin C}{\sin A} = 58.232 \times \frac{\sin 36°00′08.33″}{\sin 54°38′28.33″} = 41.972\ \text{m} \end{aligned}$$

Answer: corrected angles A = 54°38′28.33″, B = 89°21′23.33″, C = 36°00′08.33″; span of the bridge AB = 41.972 m (about 41.97 m, since the instrument reads only to 1′).