In the adjacent figure, ABCD is a parallelogram and line segments… - Vidyadip

Question

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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Answer

In $\triangle\text{AD}$ and $\triangle\text{CBF},$
we have $\text{AD}=\text{BC},\angle\text{B}=\angle\text{D}$ and $\angle\text{DAE}=\angle\text{BCF}$
$\because\angle\text{A}=\angle\text{C}$
$\Rightarrow\frac{1}{2}\angle\text{A}=\frac{1}{2}\angle\text{C}$
$\Rightarrow\angle\text{DAE}=\angle\text{BCF}$
$\therefore\triangle\text{ADE}\cong\triangle\text{CBF}$ And therefore,$\text{CD}-\text{DE}=\text{AB}-\text{BF}$ So, $CE = AF$
$\therefore$ $AECF$ is a parallelogram, Hence, $AE\ ||\ CF$.

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