Question

Question 176
ABCD is a parallelogram. The bisector of angle A intersects CD at X and bisector of angle C intersects AB at Y. Is AXCY a parallelogram? Give reason.

Answer 1

Given, ABCD is a parallelogram.
So, $\angle A=\angle C$ [$\because$ opposite angles of a parallelogram are equal]

$\therefore \frac{\angle A}{2}=\frac{\angle C}{2}$ [dividing both the sides by 2]
$\angle 1=\angle 2$
But $\angle 2=\angle 3$ [$\because AB\parallel CD$ and CY is the transversal]
$\therefore \angle 1=\angle 3$
But they are pair of corresponding angles.
$\therefore AX\parallel YC$ ..... (i)
$AY\parallel XC$ [$\because AB\parallel DC$] ...... (ii)
From Eqs. (i) and (ii), we get
AXCY is a parallelogram.