Prove that if two angles of one triangle are equal to two angles of another triangle, then the ratio of the sides including these angles are equal.

We are given two triangles, $\triangle PQR$ and $\triangle STU$.
We are also given that two pairs of corresponding angles are equal:

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

If two angles of one triangle are equal to two angles of another triangle, then the third angles must also be equal (since the sum of angles in a triangle is $180^\circ$). This means the triangles are similar.

In $\triangle PQR$ and $\triangle STU$:

1. $\hat{P} = \hat{S}$ (Given)
2. $\hat{Q} = \hat{T}$ (Given)
3. $\hat{R} = \hat{U}$ (Sum of angles in a triangle)

Therefore, $\triangle PQR \sim \triangle STU$ (Angle-Angle-Angle similarity criterion).

When two triangles are similar, the ratio of their corresponding sides is equal.
The corresponding sides are:

• PQ corresponds to ST (opposite $\hat{R}$ and $\hat{U}$)
• PR corresponds to SU (opposite $\hat{Q}$ and $\hat{T}$)
• QR corresponds to TU (opposite $\hat{P}$ and $\hat{S}$)

So, we can write the ratios of corresponding sides:

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

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

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