Question

in the adjoining figure, ABCD is a parallelogram in which line segments AE and CF bisect the the angles angle A and angle C respectively. show that AE||CF.​

Answer 1

According to question, ∠ ∠A =  ∠ ∠C (Opposite angles) Line segments AE and CF bisect the  ∠ ∠A  and  ∠ ∠C means, √ 100 100 =  1 2 12 ∠ ∠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.Read more on Sarthaks.com - https://www.sarthaks.com/1132181/adjacent-figure-abcd-parallelogram-line-segments-bisect-angles-crespectively-show-that