Question

If the bisectors of two adjacent angles form a right angle then prove that their non common arms are in same straight line

Answer 1

Refer to the attached image.

Let $\angle ABD$ and $\angle BDC$ be two adjacent angles.

Let BE be the bisector of angle ABD and BF be the bisector of angle BDC.

And angle EBF = 90 degree (Given)

To prove:  That the non common arms are in same straight line.

That is to prove that the arms AB and BC are in straight line.

Proof:

Since, angle ABD and angle BDC are the adjacent angles.

The adjacent angles forms linear pair, that is the sum of these angles is 180 degrees.

angle ABD + angle BDC = 180 degrees.

Therefore, AB and BC are in straight line.

Hence, the non common arms are in same straight line.

Answer 2

Let \angle ABD∠ABD and \angle BDC∠BDC be two adjacent angles.

Let BE be the bisector of angle ABD and BF be the bisector of angle BDC.

And angle EBF = 90 degree (Given)

To prove: That the non common arms are in same straight line.

That is to prove that the arms AB and BC are in straight line.

Proof:

Since, angle ABD and angle BDC are the adjacent angles.

The adjacent angles forms linear pair, that is the sum of these angles is 180 degrees.

angle ABD + angle BDC = 180 degrees.

Therefore, AB and BC are in straight line.

Hence, the non common arms are in same straight line.