Question

ABCD is a quadrilateral in which all four sides are equal show that both pairs of opposite sides are parallel​

Answer 1

Answer:

Answer is in the image

Step-by-step explanation:

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Answer 2

Answer:

If one pair of opposite sides of a quadrilateral is equal and parallel, then the quadrilateral is a parallelogram.

Theorem: In a parallelogram, opposite sides are equal.

Step-by-step explanation:

Statement: In a parallelogram, opposite sides are equal.

Proof: 

First, we suppose that ABCD is a parallelogram. Compare ΔABC ΔABC and ΔCDA ΔCDA:

1. AC = AC (common side)

2. ∠1∠1 = ∠4∠4 (alternate interior angles)

3. ∠2∠2 = ∠3∠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 ΔABC and ΔCDA 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∠1 = ∠4∠4  AB || CD

2. ∠2∠2 = ∠3∠3     AD || BC

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