# In figure, if PQRS is a parallelogram and AB || PS, then prove that OC || SR. - Mathematics

Sum

In figure, if PQRS is a parallelogram and AB || PS, then prove that OC || SR.

#### Solution

Given,

PQRS is a parallelogram,

Therefore, PQ || SR and PS || QR.

Also given, AB || PS.

To prove:

OC || SR

From ∆OPS and OAB,

PS||AB

∠POS = ∠AOB ......[Common angle]

∠OSP = ∠OBA   ......[Corresponding angles]

∆OPS ∼ ∆OAB   ......[By AAA similarity criteria]

Then,

Using basic proportionality theorem,

We get,

(PS)/(AB) = (OS)/(OB)   ......(i)

From ∆CQR and ∆CAB,

QR || PS || AB

∠QCR = ∠ACB   ......[Common angle]

∠CRQ = ∠CBA   .......[Corresponding angles]

∆CQR ∼ ∆CAB

Then, by basic proportionality theorem

(QR)/(AB) = (CR)/(CB)

(PC)/(AB) = (CR)/(CB)  ......(ii)

[PS ≅ QR Since, PQRS is a parallelogram,]

From Equation (i) and (ii),

(OS)/(OB) = (CR)/(CB)

(OB)/(OS) = (CB)/(CR)

Subtracting 1 from L.H.S and R.H.S, we get,

(OB)/(OS) - 1 = (CB)/(CR) - 1

(OB - OS)/(OS) = ((CB - CR))/(CR)

(BS)/(OS) = (BR)/(CR)

SR || OC  ......[By converse of basic proportionality theorem]

Hence proved.

Concept: Basic Proportionality Theorem (Thales Theorem)
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#### APPEARS IN

NCERT Mathematics Exemplar Class 10
Chapter 6 Triangles
Exercise 6.4 | Q 4 | Page 73

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