1. **Problem Statement:** We have a parallelogram ABCD with sides given as: - AB = $3x - 5$ cm - BC = $2y - 7$ cm - CD = $x + 7$ cm - AD = $y + 3$ cm We need to find the values of $x$ and $y$, the lengths of AB and AD, and the perimeter of the parallelogram. 2. **Properties of a Parallelogram:** Opposite sides of a parallelogram are equal in length. Therefore: - AB = CD - BC = AD 3. **Set up equations using these properties:** - From AB = CD: $$3x - 5 = x + 7$$ - From BC = AD: $$2y - 7 = y + 3$$ 4. **Solve for $x$:** $$3x - 5 = x + 7$$ Subtract $x$ from both sides: $$3x - x - 5 = 7$$ $$2x - 5 = 7$$ Add 5 to both sides: $$2x = 12$$ Divide both sides by 2: $$x = 6$$ 5. **Solve for $y$:** $$2y - 7 = y + 3$$ Subtract $y$ from both sides: $$2y - y - 7 = 3$$ $$y - 7 = 3$$ Add 7 to both sides: $$y = 10$$ 6. **Find the length of AB:** Substitute $x=6$ into AB: $$AB = 3(6) - 5 = 18 - 5 = 13 \text{ cm}$$ 7. **Find the length of AD:** Substitute $y=10$ into AD: $$AD = 10 + 3 = 13 \text{ cm}$$ 8. **Find the perimeter of parallelogram ABCD:** Perimeter $P = 2(AB + AD)$ $$P = 2(13 + 13) = 2(26) = 52 \text{ cm}$$ **Final answers:** - $x = 6$ - $y = 10$ - $AB = 13$ cm - $AD = 13$ cm - Perimeter $= 52$ cm