Prove that the diagonals of rectangle LMNO are congruent. You can write a two-column proof or a paragraph proof, but be sure to include a reason for each of your statements.

Answer 1

To prove that the diagonals of rectangle LMNO are congruent, we can use a two-column proof format.

Two-Column Proof

| Statement | Reason | |------------------------------------------------|---------------------------------------------| | 1. Let LMNO be a rectangle. | Definition of a rectangle. | | 2. Define the vertices: L, M, N, O such that L is opposite to N and M is opposite to O. | Definition of rectangle vertices. | | 3. The diagonals are LN and MO. | Diagonals are line segments connecting opposite vertices. | | 4. In a rectangle, opposite sides are equal (LM = ON and LN = MO). | Property of rectangles (opposite sides are equal). | | 5. Triangle LMO and triangle NMO are formed by the diagonals and sides of the rectangle. | By drawing diagonals LN and MO, we create triangles. | | 6. LM = ON (by property of rectangles). | Opposite sides of rectangle are equal. | | 7. OM = LN (by property of rectangles). | Opposite sides of rectangle are equal. | | 8. MN = LN (common side of both triangles). | Common side property. | | 9. Triangles LMO and NMO are congruent (by SAS). | Side-Angle-Side congruence criterion. | | 10. If two triangles are congruent, then their corresponding sides are equal. | Definition of congruent triangles. | | 11. Therefore, LN = MO. | Corresponding parts of congruent triangles are equal (CPCTC). |

Conclusion

We have shown through our proof that the diagonals of rectangle LMNO (LN and MO) are congruent using the properties of rectangles and the criteria for triangle congruence. Therefore, \( LN \cong MO \).