Statements | Reasons |
ABCD is a parallelogram | Given |
AC is a diagonal given | Given |
∠BCA = ∠DAC | Alternate interior angles |
∠BAC = ∠DCA | Alternate interior angles |
AC = CA | Reflexive (common side) |
ΔABC = ΔCDA | ASA postulate |
Statements of parallelogram and its theorems 1) In a parallelogram, opposite sides are equal. 2) If each pair of opposite sides of a quadrilateral is equal then it is a parallelogram. 3) In a parallelogram, opposite angles are equal. 4) If in a quadrilateral, each pair of opposite angles is equal then it is a parallelogram. 5) The diagonals of a parallelogram bisect each other. 6) If the diagonals of a quadrilateral bisect each other, then it is a parallelogram. 7) A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel. |
Statements | Reasons |
1) ABCD is a parallelogram | 1) Given |
2) DP = BQ | 2) Given |
3) AD = BC | 3) Properties of parallelogram |
4) ∠ADP = ∠CBQ | 4) Alternate interior angles |
5) ΔAPD =ΔCQB | 5) SAS Postulate |
6) AP = CQ | 6) CPCTC |
7) AB = CD | 7) Properties of parallelogram |
8) BQ = DP | 8) Given |
9) ∠ABQ = ∠CDP | 9) Alternate interior angles |
10) ΔAQB = ΔCPD | 10) SAS Postulate |
11) AQ = PC | 11) CPCTC |
12) APCQ is a parallelogram | 12) Two pairs of opposite sides of ≅then it is parallelogram. From (6) and (11) |
Statements | Reasons |
1) ABCD is a parallelogram | 1) Given |
2) AP and CQ are perpendiculars | 2) Given |
3) ∠APB = ∠CQD | 3) Definition of perpendiculars.Each of measure 90^{0} |
4) ∠ABP = ∠CDQ | 4) Alternate interior angles |
5) AB = CD | 5) Properties of parallelogram |
6) ΔAPB = ΔCQD | 6) AAS |
7) AP = CQ | 7) CPCTC |