In below fig, OP, OQ, OR and OS arc four rays. Prove that:

∠POQ + ∠QOR + ∠SOR + ∠POS = 360°

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#### Solution

Given that

*OP*, *OQ*, *OR *and *O**S *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

*`∠`T**OP *+ `∠`*POQ *= 180° .......(1)

Similarly, ray OS stands on the line *TOQ*

*∴`∠`T**O**S *+ *`∠`**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°

*⇒ `∠`**T**O**P *+ *`∠`**T**O**S *= *`∠`**P**O**S*

∴ (4) becomes

*`∠`POQ *+ *`∠`**QOR *+ *`∠`**SOR *+ *`∠`**POS *= 360°

Concept: Pairs of Angles

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