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Question: In the given figure ABCD is a quadrilateral in which AB = AD. The bisector of \(\angle \)BAC and \(\...

In the given figure ABCD is a quadrilateral in which AB = AD. The bisector of \angle BAC and \angle CAD intersect the sides BC and CD at the points E and F respectively. Prove that EF || BD.

Explanation

Solution

Hint: First, join AC, BD and EF. Now, in triangle CAB, AE is the bisector of \angle BAC. Use the angle bisector theorem to get ACAB=CEBE\dfrac{AC}{AB}=\dfrac{CE}{BE}. Similarly, from triangle CAD, you get ACAD=CFDF\dfrac{AC}{AD}=\dfrac{CF}{DF}. Substitute AB = AD to get ACAB=CFDF\dfrac{AC}{AB}=\dfrac{CF}{DF}. Equate the previous equation with this one to get CEEB=CFFD\dfrac{CE}{EB}=\dfrac{CF}{FD}. Now, use the converse of Intercept Theorem to get the final answer.

Complete step by step answer:
In this question, we are given that ABCD is a quadrilateral in which AB = AD. The bisector of \angle BAC and\angle CAD intersect the sides BC and CD at the points E and F respectively.
We need to prove that EF || BD.
Construction: Join AC, BD and EF.
In triangle CAB, AE is the bisector of \angle BAC.
Now, we will use the angle bisector theorem on triangle CAB.
The Angle Bisector Theorem says that an angle bisector of a triangle will divide the opposite side into two segments that are proportional to the other two sides of the triangle.
Using the above theorem, we will get the following:
ACAB=CEBE\dfrac{AC}{AB}=\dfrac{CE}{BE} …(1)
Similarly, in triangle CAD, AE is the bisector of \angle DAC.
Now, we will use the angle bisector theorem on triangle CAB.
Using the angle bisector theorem, we will get the following:
ACAD=CFDF\dfrac{AC}{AD}=\dfrac{CF}{DF}
Now, we are given that AB = AD. Substituting this in the above equation, we will get the following:
ACAB=CFDF\dfrac{AC}{AB}=\dfrac{CF}{DF} …(2)
From equations (1) and (2), we will get the following:
CEEB=CFFD\dfrac{CE}{EB}=\dfrac{CF}{FD}
Thus, in triangle CBD, E and F divide the sides CB and CD respectively in the same ratio. Therefore, by the converse of Intercept Theorem, we have EF ∣∣ BD.
Hence proved.

Note: In this question, it is very important to know the angle bisector theorem. It is also important to know the intercept theorem. The intercept theorem states if a transversal makes equal intercepts on three or more parallel lines, then any other line cutting them will also make equal intercepts.