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Question

In the adjacent figure OP,OQ,OR and OS are four rays. Prove that
POQ+QOR+SOR+POS=360o.
570649_027846f087764de28dbb12063a0a1f63.png

Solution
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In the given figure, you need to draw opposite ray to any of the rays OP,OQ,OR or OS

Draw ray OT so that −−TOQ is a line. Now, ray OP stands on line TQ.

Therefore, TOP+POQ=180o .... (1) (Linear pair axiom)

Similarly, ray OS stands on line TQ.

Therefore, TOS+SOQ=180o .... (2) (Why?)

But SOQ=SOR+QOR

So, (2) becomes
TOS+SOR+QOR=180o ..... (3)

Now, adding (1) and (3), you get
TOP+POQ+TOS+SOR+QOR=360o ... (4)

But TOP+TOS=POS

Therefore, (4) becomes
POQ+QOR+SOR+POS=360o

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570649_027846f087764de28dbb12063a0a1f63.png
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