To set up the two-column proof that parallelogram \(ABCD\) with congruent diagonals \(DB\) and \(AC\) is a rectangle, you can organize the statements and reasons as follows:
| Statements | Reasons | |------------|---------| | 1. \(ABCD\) is a parallelogram. | Given | | 2. \(DB \cong AC\) | Given | | 3. \(DA \cong CB\) | Opposite sides of a parallelogram are congruent. | | 4. \(AB \cong BA\) | Reflexive Property of Congruence | | 5. \(\triangle BAD \cong \triangle ABC\) | SSS Congruence Theorem | | 6. \(\angle BAD \cong \angle ABC\) | CPCTC | | 7. \(m\angle BAD = m\angle ABC\) | Definition of congruence | | 8. \(m\angle BAD + m\angle ABC = 180^\circ\) | Consecutive angles of a parallelogram are supplementary. | | 9. \(m\angle BAD + m\angle BAD = 180^\circ\) | Substitution | | 10. \(2m\angle BAD = 180^\circ\) | Combine like terms | | 11. \(m\angle BAD = 90^\circ\) | Division Property of Equality | | 12. \(m\angle ABC = 90^\circ\) | Transitive Property of Equality | | 13. \(m\angle ADC = m\angle DAB = 90^\circ\) | Opposite angles of a parallelogram are congruent. | | 14. \(\angle BAD, \angle DCB, \angle ABC, \text{ and } \angle ADC\) are right angles | Definition of a right angle | | 15. \(ABCD\) is a rectangle | Definition of a rectangle |
Please fill in response areas where specific statements or reasons were indicated, using the provided responses. The layout should guide you to ascertain the correct sequence in forming the two-column proof.