1. The problem asks for the conditions that guarantee a quadrilateral is a parallelogram. 2. One key property of parallelograms is that their opposite sides are equal and parallel. 3. Given the quadrilateral PERS with diagonals intersecting at S, and marked equal segments PE \cong AR and PA \cong ER, we analyze the conditions: 4. From the given, RE / P \cong LR and LE \cong LA, and PS \cong SR, we can infer that the diagonals bisect each other, which is a defining property of parallelograms. 5. Also, triangles \triangle PAR \cong \triangle REP by side-side-side congruence, implying PA \parallel ER and PA \cong ER. 6. Therefore, the conditions that guarantee the quadrilateral is a parallelogram are: - Opposite sides are equal: PA \cong ER and PE \cong AR. - Opposite sides are parallel: PA \parallel ER. - Diagonals bisect each other: PS \cong SR. 7. Now, for the quadrilateral CUTE, given \angle C = 110^\circ, we find the measures of \angle T, \angle U, and \angle E. 8. Since the sum of interior angles in a quadrilateral is 360^\circ, we use: $$\angle C + \angle U + \angle T + \angle E = 360^\circ$$ 9. Given \angle C = 110^\circ, and assuming \angle U, \angle T, and \angle E are unknown, we need more information to find each angle. 10. For the lengths, given CE = 2x + 7 and UT = 4x - 3, and CE = 12, we solve: $$2x + 7 = 12 \Rightarrow 2x = 5 \Rightarrow x = 2.5$$ 11. Then, $$UT = 4(2.5) - 3 = 10 - 3 = 7$$ 12. For CE = 12 and CU = 8, to find UT and ET, more information about the quadrilateral or relationships between sides is needed. 13. For angles \angle C = 3x + 25^\circ and \angle T = 4x + 5^\circ, since opposite angles in a parallelogram are equal, or adjacent angles are supplementary, we can set equations accordingly. 14. For example, if \angle C and \angle T are supplementary: $$3x + 25 + 4x + 5 = 180 \Rightarrow 7x + 30 = 180 \Rightarrow 7x = 150 \Rightarrow x = \frac{150}{7} \approx 21.43$$ 15. Then, $$\angle C = 3(21.43) + 25 = 64.29 + 25 = 89.29^\circ$$ $$\angle T = 4(21.43) + 5 = 85.72 + 5 = 90.72^\circ$$ 16. Similarly, for \angle C = 3x + 25^\circ and \angle U = x + 65^\circ, if they are supplementary: $$3x + 25 + x + 65 = 180 \Rightarrow 4x + 90 = 180 \Rightarrow 4x = 90 \Rightarrow x = 22.5$$ 17. Then, $$\angle C = 3(22.5) + 25 = 67.5 + 25 = 92.5^\circ$$ $$\angle U = 22.5 + 65 = 87.5^\circ$$ 18. Using these values, lengths LU, LT, and LE can be found if additional side length relationships are provided. Final answers: - Conditions for parallelogram: Opposite sides equal and parallel, diagonals bisect each other. - For CUTE, angles and lengths depend on given expressions and solving for x as shown.