Question

in a parallelogram opposite side are equal prove​

Answer 1

Answer:

$\huge \red{ \underline{ \underline{ \bold \pink{Theorem - }}}}$In a parallelogram, opposite sides are equal. Conversely, if the opposite sides in a quadrilateral are equal, then it is a parallelogram.

$\huge \pink{ \underline{ \underline{ \bold{PROOF - }}}}$

Refer to the above attachment.

Now, , we suppose that ABCD is a parallelogram. Compare

ΔABC and

ΔCDA

:

1. AC = AC (common side)

2.

∠1 = ∠4 (alternate interior angles)

3.

∠2= ∠3 (alternate interior angles)

Thus, by the ASA criterion, the two triangles are congruent, which means that the corresponding sides must be equal. Thus, AB = CD and AD = BC.

Now, we will prove the converse of this. Suppose that ABCD is a quadrilateral in which AB= CD and AD = BC. Compare

ΔABC and ΔCDA

once again:

1. AC = AC (common side)

2. AB = CD (given)

3. AD = BC (given)

Thus, by the SSS criterion, the two triangles are congruent, which means that the corresponding angles are equal:

1.

∠1 = ∠4

AB || CD

2.

∠2 = ∠3

AD || BC