# <span lang="EN-US

Given: OA=OB and OP=OQ

To prove: PX=QX and AX=BX

Proof:

In ∆OAQ and ∆OPB, we have

OA = OB …given

O = O …common angle

OQ = OP … given

Thus by SAS property of congruence,

∆OAP ∆OPB

Hence, we know that, corresponding parts of the congruent triangles are equal

OBP = OAQ …(1)

Thus, in ∆BXQ and ∆PXA, we have,

BQ = OB – OQ

And PA = OA – OP

But OP = OQ

And OA = OB …given

Hence, we have, BQ = PA …(2)

Now consider ∆BXQ and ∆PXA,

BXQ = PXA … vertically opposite angles

OBP = OAQ …from 1

BQ = PA … from 2

Thus by AAS property of congruence,

∆BXQ PXA

Hence, we know that, corresponding parts of the congruent triangles are equal

PX = QX

And AX = BX

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