Name: Understanding Vertically Opposite Angles
Uploaded: 2019-07-19
Duration: 6 min 7 s
Description: Did you know that many examples of vertically opposite angles can be seen in our daily lives? A railroad crossing sign for example. So how do we define this...

Question

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.

Accepted Answer

Let us consider four rays, $O P, O Q, O R$ and $O S$ 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 $O$ .

We know that opposite angles are always equal.

So, $∠ P O Q = ∠ S O R$ and $∠ P O S = ∠ Q O R$

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

So,

$∠ P O Q + ∠ S O R + ∠ P O S + ∠ Q O R = 360 °$

$\Rightarrow 2 ∠ P O Q + 2 ∠ P O S = 360 °$

$\Rightarrow 2 (∠ P O Q + ∠ P O S) = 360 °$

$\Rightarrow ∠ P O Q + ∠ P O S = 180 °$

The above equation suggests that sum of any two side angles is $180 °$ .

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.

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.

Important Questions on Concept of Vertically Opposite Angles

Learn and Prepare for any exam you want !