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In Figure altitudes AD and CE of triangl...

In Figure altitudes `AD` and `CE` of `triangle ABC` intersect each other at the point P. Show that: (i) `DeltaA E P~ DeltaC D P` (ii) `DeltaA B D ~ DeltaC B E` (iii) `DeltaA E P~ DeltaA D B` (iv) `DeltaP D C~ DeltaB E C`

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(i) in figure, `angleAEP = angleCDP " " (each = 90^(@)`
and ` angleAPE = angleCPD` ( vertically opposite angles)
`Rightarrow " " triangleAEP ~ triangleCDP` (by AAA similarity critertion)
Hence proved
`(ii) figure , angleADB= angleCEB " " (each = 90^(@))`
`and " " angleABD= angleCBE` ( by AAA similarity criterion) Hence proved.
( iii) in figure, ` angleAEP = angleADB " " (each 90^(@))`
` and anglePAE = angleBAD` (common angle)
`Rightarrow " " triangleAEP ~ triangleADB` ( by AAA similarity criterion ) Hence proved.
(iv) in figure , `anglePDC = angleBEC ( each = 90^(@))`
and ` anglePCD = angleBCE` ( common angle)
` Rightarrow " " trianglePDC ~ triangleBEC` ( by AAA similarity criterion) Hence proved.
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