Answer:
1. Parallelogram ABCD with diagonals AC and BD
DEFINITION OF PARALLELOGRAM.
2.B C || AD DEFINITION OF PARALLELOGRAM.
3.CBD= ADB and BCA = DAC (ASA) property
4.BC = AB CPCTC ( Corresponding parts of a congruent triangle)
5. AED = CEB (ASA) property
6. DE=BE and AE= CE : DEFINITION OF PARALLELOGRAM.
Explanation:
Here, given: ABCD is a parallelogram.
AC and BD are diagonals.
Now, consider the given statements one by one:
1. Parallelogram ABCD with diagonals AC and BD.
This information is already by DEFINITION OF PARALLELOGRAM.
2.BC|| AD
Definition of a parallelogram as Opposite sides in the parallelogram are EQUAL and PARALLEL.
3. ΔCBD = ΔADB and ΔBCA = ΔDAC
Now, here: as the opposites are parallel.
So, they form a pair of ALTERNATE angles.
Also, CB is a common segment in both triangle. and AC is common segment in second pair of triangles. ( REFLEXIVE Property of congruence)
Hence, ΔCBD = ΔADB and ΔBCA = ΔDAC by ANGLE SIDE SINGLE (ASA) property
4.BC = AB
CPCTC ( Corresponding parts of a congruent triangle)
5. ΔAED = ΔCEB
Now, here: as the opposites are parallel.
So, they form a pair of ALTERNATE angles.
Also, opposites are equal. ⇒ AD = BC
⇒ ΔAED = ΔCEB by ANGLE SIDE SINGLE (ASA) property
6. DE = BE and AE = CE
As, given AC and BD are diagonals.
Also, in a parallelogram, DIAGONALS BISECT EACH OTHER.
⇒DE = BE and AE = CE by DEFINITION OF PARALLELOGRAM.