Drag and drop the statements and reasons into their correct locations in the two-column proof that parallelogram ABCD with congruent diagonals DB and AC is a rectangle.

Question

Statement                                                                          Reason
1. ABCD is a parallelogram.                                              Given
2. DB AC                                                                           Given
3. DA CB                                                                           Opposite sides of a parallelogram are congruent.
4. AB BA                                                                            Reflexive Property of Congruence
5. ABAD AABC                                                                  SSS Congruence Theorem
6. ZBAD ZABC                                                                   _____
7. mBAD = m/ABC                                                           Definition of congruence
8. mBAD+m/ABC=180°                                                   Consecutive angles of a parallelogram are supplementary
9. mBAD+m/BAD= 180°                                                  ____
10. 2m/BAD = 180°                                                           Combine like terms
11. ____                                                                            Division Property of Equality
12. mABC = 90°                                                               Transitive Property of Equality
13. mADC=m/DAB = 90°                                                  Opposite angles of a parallelogram are congruent
14. ZBAD, ZDCB, ZABC, and ZADC are right angles       Definition of a right angle
15. ABCD is a rectangle                                                    Definition of a rectangle

Substitution Property of Equality
CPCTC Theorem of Congruence
mBAD=90°
Transitive Property of Congruence
mBAD/2=90°

GPT-4o mini · Answer

Sure! Here's the completed two-column proof with the statements and reasons appropriately matched.

Proof that parallelogram ABCD with congruent diagonals DB and AC is a rectangle:

| Statement | Reason | |---------------------------------------------------------------------------|------------------------------------------------------------| | 1. ABCD is a parallelogram. | Given | | 2. DB ≅ AC | Given | | 3. DA ≅ CB | Opposite sides of a parallelogram are congruent. | | 4. AB ≅ AB | Reflexive Property of Congruence | | 5. ΔABD ≅ ΔACB | SSS Congruence Theorem | | 6. ∠BAD ≅ ∠ABC | CPCTC Theorem of Congruence | | 7. m∠BAD = m∠ABC | Definition of congruence | | 8. m∠BAD + m∠ABC = 180° | Consecutive angles of a parallelogram are supplementary | | 9. m∠BAD + m∠BAD = 180° | Substitution Property of Equality | | 10. 2m∠BAD = 180° | Combine like terms | | 11. m∠BAD = 90° | Division Property of Equality | | 12. m∠ABC = 90° | Transitive Property of Equality | | 13. m∠ADC = m∠DAB = 90° | Opposite angles of a parallelogram are congruent | | 14. ∠BAD, ∠DCB, ∠ABC, and ∠ADC are right angles | Definition of a right angle | | 15. ABCD is a rectangle | Definition of a rectangle |

This arrangement clearly lays out the logical flow from the properties of parallelograms to the conclusion that ABCD is a rectangle.