In the given figure, PQ is the diameter of the circle with the center O. If PX = XS, prove that OX//QS.
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Answer:OX//QS
Step-by-step :
construction: join O and S
Statements Reasons
i) In ∆POX and ∆XOS i)...
1) PX=SX (S) 1)given
2) OX=OX (S) 2) common side
3) PO=OS(S) 3)radii of same circle
4)∆POX and ∆XOS are congruent 4)SSS
axiom
ii)OQ=OS ii)radii of same circle
iii)Angle OSQ = angle OQS iii) base angles of isosceles triangle
Iv)angle POX=XOS iv) corresponding angles of congruent triangles
V)angle XOS=1/2 angle POS v) form iv and whole part axiom
Vi)angle PQS= 1/2 angle POS vi) Relation between inscribed angle and centre angle
Vii) angle XOS= angle PQS vii) from (v) and (vii)
Viii)angle XOS= angle OSQ(i.e. OX//SQ) viii)from (vii) and (iii) and because alternate angles are equal.
Proved.....
Step-by-step :
construction: join O and S
Statements Reasons
i) In ∆POX and ∆XOS i)...
1) PX=SX (S) 1)given
2) OX=OX (S) 2) common side
3) PO=OS(S) 3)radii of same circle
4)∆POX and ∆XOS are congruent 4)SSS
axiom
ii)OQ=OS ii)radii of same circle
iii)Angle OSQ = angle OQS iii) base angles of isosceles triangle
Iv)angle POX=XOS iv) corresponding angles of congruent triangles
V)angle XOS=1/2 angle POS v) form iv and whole part axiom
Vi)angle PQS= 1/2 angle POS vi) Relation between inscribed angle and centre angle
Vii) angle XOS= angle PQS vii) from (v) and (vii)
Viii)angle XOS= angle OSQ(i.e. OX//SQ) viii)from (vii) and (iii) and because alternate angles are equal.
Proved.....
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