If each pair of opposite sides of a quadrilateral is equal,then it is ...
Given: A quadrilateral ABCD
To Prove: ABCD is a parallelogram i.e., AB ║ DC and AD ║ BC
Construction : Join A and C
Proof : In ∆ABC and ∆CDA,
Hence, ∆ABC ≅ ∆CDA [By SSS]
But these are alternate angles and if alternate angles are equal then, the lines are parallel.
Therefore, ABCD is a parallelogram.
If each pair of opposite sides of a quadrilateral is equal,then it is ...
Introduction:
A quadrilateral is a polygon with four sides. A parallelogram is a special type of quadrilateral where opposite sides are equal and parallel. In this response, we will explore whether a quadrilateral with equal opposite sides is always a parallelogram.
Definition of a Parallelogram:
A parallelogram is a quadrilateral with opposite sides that are equal in length and parallel to each other. Additionally, the opposite angles of a parallelogram are also equal.
Proof:
To determine whether a quadrilateral with equal opposite sides is a parallelogram, we can use the properties of parallelograms to prove or disprove its classification.
Opposite Sides:
If all opposite sides of a quadrilateral are equal, we can denote them as AB, BC, CD, and DA. Let's assume AB = CD and BC = DA.
Opposite Angles:
To prove that the quadrilateral is a parallelogram, we need to show that the opposite angles are also equal.
1. Consider angle A and angle C.
- Since AB = CD and BC = DA, we can conclude that triangle ABC is congruent to triangle CDA by the Side-Side-Side (SSS) congruence criterion.
- Therefore, angle A is congruent to angle C.
2. Similarly, consider angle B and angle D.
- Since AB = CD and BC = DA, we can conclude that triangle BCD is congruent to triangle DAB by the SSS congruence criterion.
- Therefore, angle B is congruent to angle D.
Conclusion:
By proving that opposite angles are congruent, we have shown that a quadrilateral with equal opposite sides is indeed a parallelogram. Hence, the statement is true.
Summary:
In summary, a quadrilateral with equal opposite sides is a parallelogram. The proof involves demonstrating that the opposite angles are also equal by using the congruence of triangles. This property is a defining characteristic of parallelograms.
To make sure you are not studying endlessly, EduRev has designed Class 9 study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in Class 9.