Question

In Fig. 8.45, OP, OQ, OR and OS are four rays, prove that:
∠POQ + ∠QOR + ∠SOR + ∠POS = 360°

Answer 1

GIVEN : OP, OQ, OR and OS are four rays.

 

To prove : ∠POQ + ∠QOR + ∠SOR + ∠POS = 360°

Construction : From O Produce T such that TOQ is a line.

 

Proof :  

Ray OP stands on line TOQ:;

∠TOP + ∠POQ = 180° [Linear pair] ………...(1)

 

Similarly,

∠TOS + ∠SOQ = 180°  [Linear pair]  

∠TOS + (∠SOR + ∠OQR) = 180° ………. (2)

Adding eq (1) and (2), we obtain :

∠TOP + ∠POQ + ∠TOS + ∠SOR + ∠QOR = 360°

(∠TOP + ∠TOS)  + ∠POQ + ∠SOR + ∠QOR = 360°

[∠TOP + ∠TOS = ∠POS]

∠POS + ∠POQ + ∠SOR + ∠QOR = 360°

Hence, ∠POQ + ∠QOR + ∠SOR + ∠POS = 360°

HOPE THIS ANSWER WILL HELP YOU……

 

Some more questions :  

In Fig. 8.46, ray OS stand on a line POQ. Ray OR and ray OT are angle bisectors of ∠POS and ∠ respectively. If ∠POS = x, find ∠ROT.

https://brainly.in/question/15905582

 

In Fig. 8.34, rays OA, OB, OC, OD and OE have the common endpoint, O. Show, that ∠AOB +∠BOC +∠COD +∠DOE + ∠EOA = 360°.

https://brainly.in/question/15905564

Answer 2

Step-by-step explanation:

Given that

OP, OQ, OR and OS are four rays

You need to produce any of the ray OP, OQ, OR and OS backwards to a point in the figure. Let us produce ray OQ backwards to a point

T so that TOQ is a line

Ray OP stands on the TOQ

∠TOP + ∠POQ = 180° [Linear pair] ………...(1)

 

Similarly,

∠TOS + ∠SOQ = 180°  [Linear pair]  

∠TOS + (∠SOR + ∠OQR) = 180° ………. (2)

Adding eq (1) and (2), we obtain :

∠TOP + ∠POQ + ∠TOS + ∠SOR + ∠QOR = 360°

(∠TOP + ∠TOS)  + ∠POQ + ∠SOR + ∠QOR = 360°

[∠TOP + ∠TOS = ∠POS]

∠POS + ∠POQ + ∠SOR + ∠QOR = 360°

Hence, ∠POQ + ∠QOR + ∠SOR + ∠POS = 360°