In the given figure, if ∠x + ∠y = ∠p + ∠q, then prove that AOB

1.	In the given figure, if ∠x + ∠y = ∠p + ∠q, then prove that AOB is a line.
Answer» Method I: In the given figure, let us assume that AOB is a straight line ∴ ∠x + ∠y = 180° (linear pair of angles) …(i) and ∠p + ∠q = 180° (linear pair of angles) …(ii) From (i) and (ii), we get ∠x + ∠y = ∠p + ∠q Hence proved Method II: As we know that sum of the angles around a point is 360°. ∴ ∠x + ∠y + ∠p + ∠q = 360° or ∠x + ∠y + ∠x + ∠y = 360° (∵ ∠x + ∠y = ∠p + ∠q) 2 (∠x + ∠y) = 360° ⇒ ∠x + ∠y = 180° ⇒ AOB is a straight line.

In the given figure, if ∠x + ∠y = ∠p + ∠q, then prove that AOB is a line.

Answer»

Method I: In the given figure, let us assume that

AOB is a straight line

∴ ∠x + ∠y = 180°

(linear pair of angles) …(i)

and ∠p + ∠q = 180°

(linear pair of angles) …(ii)

From (i) and (ii), we get

∠x + ∠y = ∠p + ∠q

Hence proved

Method II: As we know that sum of the angles around a point is 360°.

∴ ∠x + ∠y + ∠p + ∠q = 360°

or ∠x + ∠y + ∠x + ∠y = 360°

(∵ ∠x + ∠y = ∠p + ∠q)

2 (∠x + ∠y) = 360°

⇒ ∠x + ∠y = 180°

⇒ AOB is a straight line.