In parallelogram \(PQRS\), two points \(M\) and \(L\) are taken on diagonal \(QS\) such that \(S M= QL\) (see Fig.).
Verify that :
(i) \(∆PMS ≅ ∆RLQ\)
(ii) \(PM = RL\)
(iii) \(∆PLQ ≅ ∆RMS\)
(iv) \(PL = RM\)
(v) \(PMRL\) is a parallelogram
Solution:
Given:
In parallelogram \(PQRS\), two points \(M\) and \(L\) are taken on diagonal \(QS\) such that \(S M= QL\)
(i) To Prove: \(∆PMS ≅ ∆RLQ\)
Proof:
As \(PQRS\) is a parallelogram.
\(∠PSQ =∠ RQS\) [] ------(1)
\(∠PQS = ∠RSQ\) [] ------(2)
Now, in \(∆PMS \)and \(∆RLQ\), we have
\(PS=\) [Opposite sides of a parallelogram\(PQRS\) are equal]
\(MS =\) [Given]
\(∠PSM= ∠RQL\) [alternate interior angles are equal]
Hence, \(∆PMS ≅ ∆RLQ \) [By ]
(ii) To Prove: \(PM = RL\)
Proof:
As, \(∆PMS ≅ ∆RLQ\) [from (i)]
\(PM =\) [By C.P.C.T.] ----(3)
(iii) To Prove: \(∆PLQ ≅ ∆RMS\)
Proof:
Now, in \(∆PLQ\) and \(∆RMS\), we have
\(LQ =\) [Given]
\(∠PQL =∠RSM\) [Alternate interior angles are equal]
\(PQ = RS\) [ Opposite sides of a parallelogram \(PQRS\) are equal]
Hence, \(∆PLQ≅ ∆RMS \)[By ]
(iv) To Prove: \(PL = RM\)
As, \(∆PLQ ≅ ∆RMS \) [from (iii)]
Proof:
\(PL =\) [] ------(4)
(v) To Prove: \(PMRL\) is a parallelogram
In a quadrilateral \(PMRL\),
Proof:
Opposite are equal. [From (3) and (4)]
We know that, If each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram.
Hence, \(PMRL\) is a parallelogram.