In figure 5.18 if

Given:

∠x + ∠y = ∠p + ∠q

Theory

A ray possesses 360° when its to original position.

Taking the ray OB

∠x + ∠y + ∠p + ∠q

Forms out a complete circle

Hence;

⇒ ∠x + ∠y + ∠p + ∠q = 360°

If ∠x + ∠y = ∠p + ∠q

Then

(∠x + ∠y) + (∠p + ∠q) = 360°

(∠x + ∠y) + (∠x + ∠y) = 360°

2(∠x + ∠y) = 360°

⇒ (∠x + ∠y) = = 180°

OA and OB are rays joint at O and angle possesses by them ∠ x and ∠ y sums to 180°

Then

⇒ AB is a straight line