In Figure, `a A D\ a n d\ B E` are respectively altitudes of an isosceles triangle `A B C` with `A C=B Cdot` Prove that `A E=B D`

Explanation

Given `Delta ABC` is an isosceles triangle in which AC=BC,AD and BE are respectively two altitude to sides BC and AC respectively
To Prove AE=BD
Proof in `Delta ABC " "AC=BC` [given ]
`angle ABC = angle CAB `
[angles opposite to equal sides are equal ]

`angle ABD =angle EAB `
in `Delta AEB and Delta BDA ,`
`angle AEB = angle ADB=90^(@) [" given " ,AD _|_ BC " and " BE _|_ AC`]
` angle EAB =angle ABD ` [ from E q,(i) ]
and AB=AB [common side ]
`therefore Delta AEB cong Delta BDA ` [by AAS congruence rule ]
`rArr` AE=BD [by CPCT ]