Q. 105.0( 2 Votes )

Let’s prove that internal and external bisectors of an angle are perpendicular to each other.

Answer :

Let us first draw a figure to support our proof.

Proof:



In the above figure PQ is a straight line. O is the point of intersection of line with ray RO, TO and SO.


Let POS = x and let QOS = y



Here POS is an internal angle and QOS is an external angle.


Since PQ is a straight line,


x + y = 180°


x = 180 – y ……. (i)


Let OR be the internal angle bisector.


So POR = SOR = 1/2 POS [as angle bisector divides the given angle in two halves]


In the similar manner OT is the external angle bisector.


So, we can write SOT = TOQ = 1/2 SOQ


Now when we combine, we get a common angle bisector ROT.


ROT = SOR + SOT


ROT = 1/2 POS + 1/2 QOS


= 1/2 × (POS + QOS)


= 1/2 × (x + y)


= 1/2 × (180 – y + y) ………. (From equation (i))


= 1/2 × 180


ROT = 90°


Since the angle is 90°, it is proved that internal and external bisectors of an angle are perpendicular to each other.


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