Question

Read the following statements and choose the correct option.
Assertion (A): If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
Reason (R): In a parallelogram, diagonals bisect each other.

• A. Both A and R are true, and R is the correct explanation of A.
• B. Both A and R are true, but R is not the correct explanation of A.
• C. A is true, but R is false.
• D. A is false, but R is true.

Hint:

For Assertion-Reason questions, evaluate A and R independently first. Only after confirming both are true, ask: ``Does R logically explain or justify why A is true?'' Here, R states the property of parallelograms (diagonals bisect each other), and A uses this as a characterization — so R is indeed the correct explanation of A.

Answer 1

The Correct Option is A

Concept:
In Assertion-Reason questions, we must:

• Determine independently whether A is true.

• Determine independently whether R is true.

• If both are true, determine whether R is the correct explanation for A.

Step 1: Analyse Assertion (A).

Assertion: ``If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.'' This is a standard theorem in Euclidean geometry. The proof is as follows: if the diagonals \(AC\) and \(BD\) of a quadrilateral \(ABCD\) bisect each other at point \(O\), then:

• \(AO = OC\) and \(BO = OD\) (given).

• In triangles \(\triangle AOB\) and \(\triangle COD\): \(AO = CO\), \(BO = DO\), and \(\angle AOB = \angle COD\) (vertically opposite angles). So \(\triangle AOB \cong \triangle COD\) by SAS.

• This gives \(AB = CD\) and \(\angle OAB = \angle OCD\), implying \(AB \parallel CD\).

• Similarly, \(\triangle AOD \cong \triangle COB\), giving \(AD = BC\) and \(AD \parallel BC\).

• A quadrilateral with both pairs of opposite sides equal and parallel is a parallelogram.

Assertion A is TRUE.

Step 2: Analyse Reason (R).

Reason: ``In a parallelogram, diagonals bisect each other.'' This is also a standard theorem: in any parallelogram, the two diagonals bisect each other. Proof:

• In a parallelogram \(ABCD\), \(AB \parallel CD\) and \(AB = CD\).

• In \(\triangle AOB\) and \(\triangle COD\): \(\angle OAB = \angle OCD\) (alternate interior angles), \(\angle OBA = \angle ODC\) (alternate interior angles), and \(AB = CD\). So \(\triangle AOB \cong \triangle COD\) by ASA.

• Therefore \(AO = CO\) and \(BO = DO\), i.e.
the diagonals bisect each other.

Reason R is TRUE.

Step 3: Determine if R is the correct explanation of A.
Assertion A says: ``If diagonals bisect each other $\Rightarrow$ parallelogram.''
Reason R says: ``Parallelogram $\Rightarrow$ diagonals bisect each other.'' R is the

converse direction of A. While both statements are true, R provides the converse implication that together with A establishes the

biconditional: a quadrilateral is a parallelogram if and only if its diagonals bisect each other. In this Assertion-Reason format, R is considered the

correct explanation for A because A is justified by the defining characterization of parallelograms that R states (R provides the underlying property that enables the identification).

Both A and R are true, and R is the correct explanation of A.