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.

Question

(4 points)

Answer 1

Here's a two-column proof demonstrating that the diagonals of rectangle LMNO are congruent:

| Statements | Reasons | |--------------------------------------------------------|---------------------------------------------------| | 1. LMNO is a rectangle. | 1. Given. | | 2. By definition, a rectangle has four right angles. | 2. Definition of a rectangle. | | 3. The diagonals of rectangle LMNO are lines LM and ON. | 3. Definition of diagonals in a polygon. | | 4. In a rectangle, opposite sides are equal (LM = NO and LN = MO). | 4. Properties of rectangles. | | 5. Triangles LMO and NOL are formed by the diagonals. | 5. Diagonals intersect each other, forming two triangles. | | 6. LM = NO and LN = MO (from statement 4) and angles LMO and NOL are right angles (90 degrees). | 6. Properties of rectangles and right angles. | | 7. By the Hypotenuse-Leg (HL) theorem, triangles LMO and NOL are congruent. | 7. HL theorem states that if the hypotenuse and a leg of one triangle are equal to the hypotenuse and a leg of another triangle, the triangles are congruent. | | 8. Since triangles LMO and NOL are congruent, diagonal LM = diagonal ON. | 8. Corresponding parts of congruent triangles (CPCTC). | | 9. Therefore, the diagonals of rectangle LMNO are congruent. | 9. Conclusion drawn from congruence of segments. |

Thus, we have proven that the diagonals of rectangle LMNO are congruent.