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

Two straight lines intersect each other at a point and thus four angles are formed. Let's prove that the bisectors of these angles are two perpendicular straight lines.

Accepted Answer

Let us have,

In the above figure $P Q$ and $S R$ are two straight lines. Also, $O$ is the intersection point. $A O$ is the angle bisector of $∠ P O S$ , $O N$ is the angle bisector of $∠ S O Q$ , $B O$ is the angle bisector of $∠ Q O R$ and $M O$ is the angle bisector of $∠ P O R$ .

$∠ P O R = ∠ S O Q$ [vertically opposite angles are always equal]

Similarly, $∠ P O S = ∠ Q O R$ [vertically opposite angles are always equal]

$O R$ is perpendicular to line $P Q$ .

So $∠ P O R + ∠ Q O R = 180 °$ [as they form linear pair]

Now, $∠ M O R = ∠ M O P = \frac{1}{2} ∠ P O R$ [As $M O$ is the angle bisector]

Similarly, $∠ B O R = ∠ Q O B = \frac{1}{2} ∠ Q O R$ [as $B O$ is the angle bisector]

$∠ M O B = ∠ M O R + ∠ B O R$

$= \frac{1}{2} ∠ P O R + \frac{1}{2} ∠ Q O R$

$= \frac{1}{2} \times (∠ P O R + ∠ Q O R)$

$= \frac{1}{2} \times 180 °$

$∴ ∠ M O B = 90 °$

Also, $∠ S O Q + ∠ Q O R = 180 °$ [as they form linear pair]

$∠ N O S = ∠ N O Q = \frac{1}{2} ∠ S O Q$ [as $N O$ is the angle bisector]

Similarly, $∠ Q O B = ∠ B O R = \frac{1}{2} ∠ Q O R$ [as $B O$ is the angle bisector]

$∠ N O B = ∠ N O Q + ∠ Q O B$

$= \frac{1}{2} ∠ S O Q + \frac{1}{2} ∠ Q O R$
$= \frac{1}{2} \times (∠ S O Q + ∠ Q O R)$

$= \frac{1}{2} \times 180 °$

$∴ ∠ N O B = 90 °$

Since the angle is $90 °$ , it is proved that the bisectors of these angles are two perpendicular straight lines.

Two straight lines intersect each other at a point and thus four angles are formed. Let's prove that the bisectors of these angles are two perpendicular straight lines.

Important Questions on Concept of Vertically Opposite Angles

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