To complete the two-column proof that parallelogram ABCD with congruent diagonals DB and AC is a rectangle, we can fill in the blanks as follows:
| Statements | Reasons | |------------------------------------------------------|---------------------------------------------| | Statement 1: ∠BAD ≅ ∠ABC | Reason: CPCTC Theorem of Congruence | | Statement 2: m∠BAD + m∠BAD = 180 degrees | Reason: Supplementary Angles (or definition) | | Statement 3: m∠BAD = 90 degrees | Reason: Division Property of Equality |
This option utilizes CPCTC (Corresponding Parts of Congruent Triangles are Congruent) to establish that the angles are congruent since the diagonals of the parallelogram are congruent, which implies the formation of two congruent triangles. The second statement reflects the supplementary nature of angles in a parallelogram, indicating that their measures sum to 180 degrees. Finally, using the division property confirms that if two equal angles exist, then each must measure 90 degrees to confirm that it is indeed a rectangle.
So, for the specific options:
- Reason for Statement 1: E: CPCTC Theorem of Congruence
- Reason for Statement 2: A: Substitution Property of Equality
- Statement 3 (m∠BAD = 90 degrees): D: m∠BAD = 90 degrees
This results in a complete proof showing the properties needed to conclude that ABCD is indeed a rectangle.
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