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

According to question,


A=C (Opposite angles)


Line segments AE and CF bisect the A and C means,


=


DAE = BCF ----------(i)


Now, In triangles ADE and CBF,


AD = BC (Opposite sides)


B = D (Opposite angles)


DAE = BCF (from (i))


Therefore, Δ ADE ΔCBF (By ASA congruency)


By CPCT, DE=BF


But, CD=AB


CD - DE = AB - BF.


So, CE = AF.


Therefore, AECF is a quadrilateral having pairs of side parallel and equal,


So, AECF is a parallelogram. Hence, AE || CF.


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