ABCD is a parallelogram. AE and CF are angle bisectors of ∠DAC and ∠ACB, respectively. Show that the lines AE and CF are parallel to each other.

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Solution

In parallelogram ABCD, AB || CD and AC is a diagonal which acts as the transversal.
$∴ ∠ D A C = ∠ B C A$

Given, AE and CF are angle bisectors of $∠ D A C$ and $∠ B C A$ .
$∴ \frac{1}{2} \times ∠ D A C$ = $\frac{1}{2} \times ∠ B C A$

Thus, $∠ E A C = ∠ A C F$ which forms alternate interior angles. (AE and CF are lines and AC is a transversal)

Hence, AE and CF are parallel lines.

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