1. **Problem Statement:** Given triangles \(\triangle RST\) and \(\triangle UVW\) with \(RT = 6x - 2\), \(UW = 2x + 7\), and angles \(\angle R = \angle U\), \(\angle S = \angle V\). Find the value of \(x\) such that \(\triangle RST \cong \triangle UVW\). 2. **Concept:** To prove two triangles congruent, one common method is the Angle-Side-Angle (ASA) criterion, which states that if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent. 3. **Given:** \(\angle R = \angle U\), \(\angle S = \angle V\), and sides \(RT\) and \(UW\) opposite these angles. 4. **Apply ASA Criterion:** Since two pairs of angles are equal, the side between these angles must be equal for congruence. Thus, set \(RT = UW\): $$6x - 2 = 2x + 7$$ 5. **Solve for \(x\):** $$6x - 2 = 2x + 7$$ $$6x - 2x = 7 + 2$$ $$4x = 9$$ $$x = \frac{9}{4} = 2.25$$ 6. **Conclusion:** The value of \(x\) must be \(2.25\) to prove \(\triangle RST \cong \triangle UVW\) by ASA criterion.