Question

Statement

Answer 1

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.

Step-by-step 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.