Prove that if two pairs of corresponding angles of two triangles are equal, then the ratio of the sides including these angles is equal.

We are given two triangles, $\triangle PQR$ and $\triangle STU$.
From the problem statement, we are given that two pairs of corresponding angles are equal:

$$\hat{P} = \hat{S}$$ $$\hat{Q} = \hat{T}$$

If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. Since $\hat{P} = \hat{S}$ and $\hat{Q} = \hat{T}$, we can conclude that $\triangle PQR$ is similar to $\triangle STU$.

$$\triangle PQR \sim \triangle STU \quad (\text{AA Similarity})$$

When two triangles are similar, the ratio of their corresponding sides is equal. Based on the similarity $\triangle PQR \sim \triangle STU$, the corresponding sides are $PQ$ and $ST$, $QR$ and $TU$, and $PR$ and $SU$.
Therefore, their ratios are equal:

$$\frac{PQ}{ST} = \frac{QR}{TU} = \frac{PR}{SU}$$

From the proportionality of corresponding sides, we can directly state the required theorem:

$$\frac{PQ}{ST} = \frac{PR}{SU}$$