In the given figure, if x + y = w + z, then prove that AOB is a line.

1.	In the given figure, if x + y = w + z, then prove that AOB is a line.
Answer» Given that x + y = w + z = k say From the figure x + y + z + w = 360° (∵ Angle around a point) Also x + y = z + w ∴ x + y = z + w = 360°/2 ∴ x + y = z + w = 180° OR k + k = 360° 2k = 360° k = 360°/2 (i.e.) (x,y) and (z, w) are pairs of adjacent angles whose sum is 180°. In other words (x, y) and (z, w) are linear pair of angles ⇒ AOB is a line.

Answer»

Given that x + y = w + z = k say

From the figure

x + y + z + w = 360°

(∵ Angle around a point)

Also x + y = z + w

∴ x + y = z + w = 360°/2

∴ x + y = z + w = 180°

k + k = 360°

2k = 360°

k = 360°/2

(i.e.) (x,y) and (z, w) are pairs of adjacent angles whose sum is 180°.

In other words (x, y) and (z, w) are linear pair of angles ⇒ AOB is a line.

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