Math, asked by ylp200947, 10 months ago

In quadrilateral ABCD, side AB congruents to side AD. bisector of Angle bac cuts side BC At E and bisector of angle DAC cuts side DC at F. Prove that segEF parallel segBD.

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Answers

Answered by bhagyashreechowdhury
10

If AB = AD and the bisector of  ∠BAC and ∠DAC intersect the sides BC and DC at the points E and F respectively, then seg EF || seg BD is proved.

Step-by-step explanation:

Referring to the figure attached below,

Consider ΔABC and ΔACD,

AE is the bisector of ∠BAC

AF is the bisector of ∠CAD

We know that according to the internal bisector theorem which states that the angle bisector of a triangle divides the opposite sides in the ratio of sides consisting of the angles

AC/AB = CE/BE …….. (i)

And

AC/AD = CF/FD

AC/AB = CF/FD ……. [given side AB = side AD] …… (ii)

From eq. (i) & (ii), we get

CE/BE = CF/FD …. (iii)

Now,  

In ΔBCD we have -  

CE/BE = CF/FD ….. [from eq. (iii)]

We know that according to the converse of BPT theorem, if a line divides any two sides of a triangle in the same ratio, then the line should be parallel to its third side.

seg EF || seg BD  

Hence Proved

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