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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.

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 According to question,

\(\angle\)A = \(\angle\)C (Opposite angles)

Line segments AE and CF bisect the \(\angle\)A  and \(\angle\)C means,

\(\sqrt{100}\) = \(\frac{1}{2}\)\(\angle\)C

∠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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