Question

ABCD is a parallelogram. If E is a midpoint of BC and AE is the bisect or of angle A, prove that AB=1/2AD

Answer 1

Given that,
angle DAE=angle BAE...........................1
and BE=EC..............................2
as it is a parallelogram ,
AD parallel BC. and here AE is a transversal.
Then, angle DAE= angle BEA...............3
as alternate interior angles.
By eq1 and. eq3 we get,
angle AEB=angle DAE = angle BAE
therefore,AB=BE..............................4
as. sides opposite to equal angles.
we know that opposite sides of a parallelogram are equal.
therefore,AD=BC
AD=BE+EC
AD=BE+BE. ( by eq2)
AD=2BE
AD/2=BE
AD/2=AB.
1/2AD=AB (by eq4)
Hence proved.