Question

Q- ᴡʀɪᴛᴇ ᴛʜᴇ ᴘʀᴏᴏꜰ ɢɪᴠᴇɴ ᴡɪᴛʜ ᴛʜᴇ ᴛʜᴇᴏʀᴇᴍ ᴏꜰ ᴄʏᴄʟɪᴄ Qᴜᴀᴅʀɪʟᴀᴛᴇʀᴀʟ

ᴛʜᴇᴏʀᴇᴍ: ᴏᴘᴘᴏꜱɪᴛᴇ ᴀɴɢᴇʟꜱ ᴏꜰ ᴀ ᴄʏᴄʟɪᴄ Qᴜᴀᴅʀɪʟᴀᴛᴇʀᴀʟ ᴀʀᴇ ꜱᴜᴘᴘʟᴇᴍᴇɴᴛᴀʀʏ


Class : 10th



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Answer 1

Answer:

Theorem: Prove the opposite angles of a Cyclic Quadrilateral is Supplementary i.e have the sum of 180°.

Given: Cyclic Quadrilateral PQRS for a Circle with centre O.

To Prove: <P + <Q =180°

<Q + <S =180°

Construction: Join OS and OQ.

Proof: Arc QPS subtends <SOQ at the centre and <SRQ at the remaining part of the circle.

Therefore, <SOQ = 2<SRQ.

Major arc SRQ subtends reflex angle SOQ at the centre and <SOQ on the remaining part of the circle.

Therefore, Reflex <SOQ= 2<SPR

2x + 2y = 360°.

Therefore, x+y = 180°.

Hence, <P + <R = 180°.

Now in Quadrilateral PQRS,

<P + <Q + <R + <S = 360° [Angle Sum Property]

(<P + <R) + <Q + <S = 360°

i.e. 180° + <Q + <S = 360°

Therefore, <Q + <S = 360° - 180°

= 180°

Hence, <Q + <S = 180°.

$\underline{Thus\;Proved.}$

[Sorry if the figure's a bit messy.. I drew it in a hurry.]

$Hope\;this\;Helps:)$

P. S. If you have any doubt s regarding this, you can ask me down in the comments. I will be Happy to Help :D

I Purple You! $\huge\bold\purple{♡}$