Step 1: Match up the vertices from the equal angles.
We are told $\angle A = \angle Q$ and $\angle B = \angle R$. Since the three angles of a triangle always add to $180^\circ$, the remaining angles must also match: $\angle C = \angle P$. So the correspondence is $A \leftrightarrow Q$, $B \leftrightarrow R$, $C \leftrightarrow P$, giving $\triangle ABC \sim \triangle QRP$.
Step 2: Use the Sine Rule inside triangle $ABC$ instead of the similarity-ratio rule.
In any triangle, a side is proportional to the sine of the angle opposite it. In $\triangle ABC$:
\[ \frac{AB}{\sin C} = \frac{AC}{\sin B} \]
Rearranging:
\[ \frac{AB}{AC} = \frac{\sin C}{\sin B} \]
Step 3: Use the Sine Rule inside triangle $QRP$.
Since $\angle Q$ is opposite side $RP$, and $\angle R$ is opposite side $QP$, in $\triangle QRP$:
\[ \frac{QR}{\sin P} = \frac{QP}{\sin R} \]
Rearranging:
\[ \frac{QR}{QP} = \frac{\sin P}{\sin R} \]
Step 4: Substitute the equal angles.
We know $\angle P = \angle C$ (from Step 1) and $\angle R = \angle B$ (given). Substituting these into the ratio from Step 3:
\[ \frac{QR}{QP} = \frac{\sin C}{\sin B} \]
Step 5: Compare with the ratio from triangle $ABC$.
From Step 2, $\frac{AB}{AC} = \frac{\sin C}{\sin B}$, which is the exact same expression found in Step 4 for $\frac{QR}{QP}$. So the two ratios must be equal.
Step 6: Final Answer.
$AB : AC$ is equal to $QR : QP$, so option (C) is correct.
\[ \boxed{AB : AC = QR : QP} \]