# In Below Fig, Op, Oq, Or and Os Arc Four Rays. Prove That: ∠Poq + ∠Qor + ∠Sor + ∠Pos = 360° - Mathematics

In below fig, OP, OQ, OR and OS arc four rays. Prove that:
∠POQ + ∠QOR + ∠SOR + ∠POS = 360°

#### Solution

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

Since ∠TOP, ∠POQ is linear pair

∠TOP + ∠POQ = 180°                  .......(1)

Similarly, ray OS stands on the line TOQ

∴∠TOS + ∠SOQ = 180°                      ..........(2)

But ∠SOQ = ∠SOR + ∠QOR

So, (2), becomes

∠TOS + ∠SOR + ∠OQR = 180°

Now, adding (1) and (3) you get

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

⇒ ∠TOP + ∠TOS = ∠POS

∴ (4) becomes

∠POQ + ∠QOR + ∠SOR + ∠POS = 360°

Concept: Pairs of Angles
Is there an error in this question or solution?

#### APPEARS IN

RD Sharma Mathematics for Class 9
Chapter 10 Lines and Angles
Exercise 10.2 | Q 14 | Page 16