ABCD is a quadrilateral in which all four sides are equal show that both pairs of opposite sides are parallel
Answers
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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Answer:
ABCD is a quadrilateral in which amm four sides are equal.Show pair of opposote sides are parallel