Question

With reference to the figure, ABCD is a parallelogram. AE and CF are angle bisectors of ∠DAC and ∠ACB respectively. What can be said about the lines AE and CF ?

• A. Always parallel
• B. Always perpendicular
• C. Can never be parallel or perpendicular
• D. Neither parallel nor perpendicular

Answer 1

The correct option is A

Always parallel

In parallelogram ABCD, AB || CD and AC is a diagonal which acts as the transversal.
$\therefore \angle DAC=\angle BCA$

Given, AE and CF are angle bisectors of $\angle DAC$ and $\angle BCA$.
$\therefore \frac{1}{2}\times \angle DAC$ = $\frac{1}{2}\times \angle BCA$

Thus, $\angle EAC=\angle ACF$ which forms alternate interior angles. (AE and CF are lines and AC is a transversal)

Hence, AE and CF are parallel lines.