1. **State the problem:** We are given the quadrilaterals ARST and AUVW with the following side lengths: - RT = 6x - 2 - UW = 2x + 7 - ZR = 2U - S = 2V We need to find the value of $x$ such that ARST is congruent to AUVW. 2. **Analyze the given information:** To prove that ARST is congruent to AUVW, corresponding sides must be equal. 3. **Set corresponding sides equal:** Since RT corresponds to UW, set: $$6x - 2 = 2x + 7$$ 4. **Solve for $x$:** $$6x - 2 = 2x + 7$$ Subtract $2x$ from both sides: $$6x - 2x - 2 = 7$$ $$4x - 2 = 7$$ Add 2 to both sides: $$4x = 9$$ Divide both sides by 4: $$x = \frac{9}{4} = 2.25$$ 5. **Check other conditions:** The problem states $ZR = 2U$ and $S = 2V$, but without numerical values for $U$ and $V$, we cannot use these to find $x$. The key is the equality of RT and UW. **Final answer:** $$x = 2.25$$