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In the figure, the circles with centers ...

In the figure, the circles with centers P and Q touch each other at R.A line passing through R meets the circles at A and B respectively. Prove that
seg AP || seg BQ

Text Solution

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`P-R-Q " ""…….."(" By theorem of touching circles ")`
In `triangle PAR`,
`seg PA cong seg PR " " "……."(" Radii of the same circle ")`
`:./_PAR cong /_PRA " " "……."(" Isosceles triangle theorem ")"…….(i)"`
In `triangle QRB`,
`seg QR cong seg QB " " "……."(" Radii of the same circle ")`
`:./_QRB cong /_QBR " " "……."(" Isosceles triangle theorem ")"…….(ii)"`
`/_PRA cong /_QRB " " "........"(" Vertically opposite angles ")".........(iii)"`
`:.` From (i),(ii) and (iii), we get ,
`/_PAR cong /_QBR`
i.e.`/_PAB cong /_QBA " " "..........."( A-R-B)`
`:. seg AP || seg BQ " " "......"(" Alternate angles test for parallel lines ")`
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