S1. Parallelogram ABCD with diagonal AC - R1. Given
S2. AB ≅ CD and AD ≅ BC - R2. Definition of Parallelogram
S3. ∠1 ≅ ∠2 and ∠3 ≅ ∠4 - R3. Opposite Angles of Parallelogram are Congruent
S4. AC ≅ AC - R4. Reflexive Property of Congruence
S5. ΔABC ≅ ΔCDA - R5. ASA Postulate (Angle-Side-Angle), since we have two angles and the included side between them congruent.
S6. AB ≅ CD and AD ≅ BC - R6. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
This completes the proof for the given parallelogram
The image provided is of a parallelogram ABCD with diagonals and includes a two-column proof with some missing steps. To prove that AB is congruent to CD and AD is congruent to BC, the proof would typically follow these steps:
1. Given: Parallelogram ABCD with diagonal AC (R1: Given).
2. Definition of Parallelogram: In a parallelogram, opposite sides are congruent (R2).
3. Opposite Angles are Congruent: In a parallelogram, opposite angles are congruent (R3).
4. Reflexive Property: A segment is congruent to itself; hence, AC ≅ AC (R4: Reflexive Property of Congruence).
5. Triangles are Congruent: Triangles ABC and CDA are congruent by the ASA Postulate (R5: ASA Postulate).
6. Corresponding Parts of Congruent Triangles are Congruent (CPCTC): If two triangles are congruent, then all of their corresponding parts are congruent (R6: CPCTC).