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. △BAD≅△ABC SSS Congruence Theorem 6. ∠BAD≅∠ABC Response area 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° Response area 10. 2m∠BAD=180° Combine like terms 11. Response area 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

Answer 1

Your statement and reason sequence appears to provide a structured proof demonstrating that quadrilateral ABCD is a rectangle. Let's clarify each part and follow a logical progression. Here’s a breakdown of the statements along with the respective reasons:

1. **Given**: ABCD is a parallelogram.
2. **Given**: DB ≅ AC (Diagonals are congruent).
3. **DA ≅ CB** (Opposite sides of a parallelogram are congruent).
4. **AB ≅ BA** (Reflexive Property of Congruence).
5. **△BAD ≅ △ABC** (SSS Congruence Theorem).
6. **∠BAD ≅ ∠ABC** (Corresponding parts of congruent triangles are congruent).
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°** (Substituting m∠ABC for m∠BAD from Step 7).
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).

### Explanation
- You start with a parallelogram, which has the property that opposite sides are equal in length and opposite angles are equal.
- You confirm that the diagonals are congruent, which is a hallmark of rectangles (alongside all angles being right angles).
- You show that certain angles must be right angles, leading to the conclusion that all angles in the parallelogram are right angles.
- Finally, you conclude that because it satisfies the definition of a rectangle (a parallelogram with all right angles), ABCD is indeed a rectangle.

This proofs demonstrates the logical flow and alignment with geometric definitions and properties, affirming your statements logically lead to the conclusion.