Prove that the bisector of interior angles of a parallelogram form a rectangle.
STEP 1 : Assumptions
Assume that is a parallelogram.
Let be the point of intersection of bisectors of and , and , and , and , respectively as shown in figure.
STEP 2 : Finding the value of
From the triangle, , we can observe that bisects and bisects .
Therefore,
[Since and are the interior angles on the same side of the transversal]
Hence,
Using the angle sum property of a triangle, we can write:
Now, substitute in the above equation, we get
STEP 3 : Finding the value of and
Since, is being vertically opposite to ,
We can say
Likewise, it can be shown that or
Similarly, and .
We have proved that and .
As, both the pairs of opposite angles are equal to , we can conclude that is a rectangle.
Hence, proved.