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.
Two-Column Proof:
Statement | Reasons
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1. Rectangle LMNO is given. | Given.
2. The opposite sides of a rectangle are parallel. | Definition of a rectangle.
3. LO is parallel to MN. | Consequence of statement 2.
4. LN is parallel to MO. | Consequence of statement 2.
5. Lines that are parallel to a same line are parallel to each other. | Transitive property of parallel lines.
6. LO and MN are parallel to LN and MO respectively. | Consequence of statement 3 and 4, and statement 5.
7. LO and MN are transversals of LN and MO respectively. | Definition of transversals.
8. Alternate interior angles formed by parallel lines and a transversal are congruent. | Alternate Interior Angles Theorem.
9. Angle LON is congruent to angle MON. | Consequence of statement 6 and statement 8.
10. LN and MO intersect at point P. | Definition of intersection.
11. LO and MN intersect at point Q. | Definition of intersection.
12. Angle LNO is congruent to angle MNO. | Consequence of statement 9 and statement 10.
13. Triangle LON and triangle MON are congruent. | ASA Congruence Postulate applied to triangle LON and triangle MON (statement 9, statement 10, statement 12).
14. LP is congruent to MP. | Consequence of statement 13 (congruent corresponding parts of congruent triangles).
15. Angle ONQ is congruent to angle QNM. | Consequence of statement 9 (congruent alternate interior angles to congruent angles).
16. Triangle LON and triangle QNO are congruent | AAS Congruence Theorem applied to triangle LON and triangle QNO (statement 9, statement 12, statement 15).
17. LO is congruent to QO. | Consequence of statement 16 (congruent corresponding parts of congruent triangles).
18. The diagonals LN and MO have common endpoint O and are congruent. | Consequence of statement 14 and statement 17 (two sides and included angle are congruent in two triangles, therefore the third side, which is the diagonal LN, and the third side, which is the diagonal MO, are congruent by the SSS Congruence Theorem).