Question

In the adjacent figure, ABCD is a parallelogram and line segments AE and CF bisect the angles A and C respectively. Show that AE || CF.

Answer 1

ABCD is a parallelogram.

AE and CF are the bisectors of $\angle Aand\angle C$ respectively.

In $△ADEand△CBF$,

AD = BC (Opposite sides of parallelogram)

$\angle D=\angle B$ (Opposite sides of parallelogram)

$\angle DAE=\angle FCB\left(\frac{1}{2}ofequalangles\right)$

$\therefore △ADE\equiv △CBF$ (S.A.A axiom of congruency)

$\therefore DE=AB$

But CD = AB (Opposite sides of parallelogram)

$\therefore CD-DE=AB-BF$

$\Rightarrow EC=AF$

and AB|| CD

$\therefore AFCE$ is a parallelogram

$\therefore$ AE || CF.