In the diagram, ABC and ACD are triangles such that angle ACB = angle ADC.

1. Identify the given equal angles

Given that
$\angle ACB = \angle ADC$.

2. Identify the second pair of equal angles and conclude similarity

Since $A$, $B$ and $D$ are collinear,
$\angle BAC = \angle CAD$.

Therefore, by AA similarity,
$\triangle ABC \sim \triangle ACD$.

1. Find the length of the corresponding side $AD$.

Since $B$ lies on $AD$,
$AD = AB + BD = 6.5 + 5.5 = 12\text{ cm}.$

2. Set up the proportion using corresponding sides of the similar triangles.

From $\triangle ABC \sim \triangle ACD$,
$AB \leftrightarrow AC, \qquad AC \leftrightarrow AD.$
Hence
$\frac{AB}{AC}=\frac{AC}{AD}.$

3. Substitute the known lengths and solve for $AC$.

$\frac{6.5}{AC}=\frac{AC}{12}$
$AC^2 = 6.5 \times 12 = 78$
$AC = \sqrt{78} = 8.831\ldots$
Therefore, to 3 significant figures,
$AC = 8.83\text{ cm}.$