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

Question

Philoid · Accepted Answer

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.