Four rays meet at a point in such a way that measurement of opposite angles are equal. Let’s prove that two straight lines are formed by those four rays.

Let us consider four rays, OP, OQ, OR and OS meet at a common point.

Now in these rays the common point is O. Hence all the rays meet at a common point O.

Now in the above figure rays meet at a common point O.

It is given that opposite angles are always equal.

In the above given figure, let us understand the pairs of vertically opposite angle.

∠POQ and ∠SOR are lying opposite to each other [in common language lying face to face to each other].

So, first pair of vertically opposite angles is ∠POQ and ∠SOR.

∠POS and ∠QOR are also lying opposite to each other.

Hence ∠POS and ∠QOR forms our second pair of vertically opposite angles.

To Prove: Four rays form two straight lines

Proof:

Let ∠POQ = ∠1

∠QOR = ∠2

∠SOR = ∠3

∠POS = ∠4.

Now when we closely look at the figure,

∠1 = ∠3 and ∠2 = ∠4.

If we add all these angles it will be sum up to 360° as the angles form a complete circle.

Let ∠1 = ∠3 = x and ∠2 = ∠4 = y

So ∠1 + ∠2 + ∠3 + ∠4 = 360°

2x + 2y = 360°

2(x + y) = 360°

x + y = 180°

The above equation suggests that sum of any two side angles is 180° that is ∠1 + ∠2 = 180°

Or ∠2 + ∠3 = 180°

Or ∠3 + ∠4 = 180°

Or ∠1 + ∠4 = 180°

The sum of two side angles is 180° which means that they form a linear pair.

Linear Pair case is possible only when two straight lines intersect at a common point.

Hence it is proved that four rays intersect to form two straight lines.