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:
- Given: ABCD is a parallelogram.
- Given: DB ≅ AC (Diagonals are congruent).
- DA ≅ CB (Opposite sides of a parallelogram are congruent).
- AB ≅ BA (Reflexive Property of Congruence).
- △BAD ≅ △ABC (SSS Congruence Theorem).
- ∠BAD ≅ ∠ABC (Corresponding parts of congruent triangles are congruent).
- m∠BAD = m∠ABC (Definition of congruence).
- m∠BAD + m∠ABC = 180° (Consecutive angles of a parallelogram are supplementary).
- m∠BAD + m∠BAD = 180° (Substituting m∠ABC for m∠BAD from Step 7).
- 2m∠BAD = 180° (Combine like terms).
- m∠BAD = 90° (Division Property of Equality).
- m∠ABC = 90° (Transitive Property of Equality).
- m∠ADC = m∠DAB = 90° (Opposite angles of a parallelogram are congruent).
- ∠BAD, ∠DCB, ∠ABC, and ∠ADC are right angles (Definition of a right angle).
- 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.
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