Question

9. Diagonal AC of parallelogram ABCD bisects A show that
i.
It bisects c also
с
ii.
ABCD is a Rhombus
A
B
Pls solve the 9 question

Answer 1

$\bold{Answer:}$

(ɪ) ʜᴇʀᴇ, ᴀʙᴄᴅ ɪꜱ ᴀ ᴘᴀʀᴀʟʟᴇʟᴏɢʀᴀᴍ ᴀɴᴅ ᴅɪᴀɢᴏɴᴀʟ ᴀᴄ ʙɪꜱᴇᴄᴛꜱ ∠ᴀ.

∴  ∠ᴅᴀᴄ=∠ʙᴀᴄ      ---- ( 1 )

ɴᴏᴡ,

ᴀʙ∥ᴅᴄ ᴀɴᴅ ᴀᴄ ᴀꜱ ᴛʀᴀᴠᴇʀꜱᴀʟ,

∴  ∠ʙᴀᴄ=∠ᴅᴄᴀ          [ ᴀʟᴛᴇʀɴᴀᴛᴇ ᴀɴɢʟᴇꜱ ]  --- ( 2 )

ᴀᴅ∥ʙᴄ ᴀɴᴅ ᴀᴀᴄ ᴀꜱ ᴛʀᴀᴠᴇʀꜱᴀʟ,

∴  ∠ᴅᴀᴄ=∠ʙᴄᴀ         [ ᴀʟᴛᴇʀɴᴀᴛᴇ ᴀɴɢʟᴇꜱ ]   --- ( 3 )

ꜰʀᴏᴍ ( 1 ), ( 2 ) ᴀɴᴅ ( 3 )

∠ᴅᴀᴄ=∠ʙᴀᴄ=∠ᴅᴄᴀ=∠ʙᴄᴀ

∴  ∠ᴅᴄᴀ=∠ʙᴄᴀ

ʜᴇɴᴄᴇ, ᴀᴄ ʙɪꜱᴇᴄᴛꜱ ∠ᴄ.

(ɪɪ)  ɪɴ △ᴀʙᴄ,

⇒  ∠ʙᴀᴄ=∠ʙᴄᴀ      [ ᴘʀᴏᴠᴇᴅ ɪɴ ᴀʙᴏᴠᴇ ]

⇒  ʙᴄ=ᴀʙ      [ ꜱɪᴅᴇꜱ ᴏᴘᴘᴏꜱɪᴛᴇ ᴛᴏ ᴇQᴜᴀʟ ᴀɴɢʟᴇꜱ ᴀʀᴇ ᴇQᴜᴀʟ ]    --- ( 1 )

⇒  ᴀʟꜱᴏ, ᴀʙ=ᴄᴅ ᴀɴᴅ ᴀᴅ=ʙᴄ     [ ᴏᴘᴘᴏꜱɪᴛᴇ ꜱɪᴅᴇꜱ ᴏꜰ ᴘᴀʀᴀʟʟᴇʟᴏɢʀᴀᴍ ᴀʀᴇ ᴇQᴜᴀʟ ]       ---- ( 2 )

ꜰʀᴏᴍ ( 1 ) ᴀɴᴅ ( 2 ),

⇒  ᴀʙ=ʙᴄ=ᴄᴅ=ᴅᴀ

ʜᴇɴᴄᴇ, ᴀʙᴄᴅ ɪꜱ ᴀ ʀʜᴏᴍʙᴜꜱ.

Answer 2

Answer:

Step-by-step explanation:

ᴀʙ∥ᴅᴄ ᴀɴᴅ ᴀᴄ ᴀꜱ ᴛʀᴀᴠᴇʀꜱᴀʟ,

∴  ∠ʙᴀᴄ=∠ᴅᴄᴀ          [ ᴀʟᴛᴇʀɴᴀᴛᴇ ᴀɴɢʟᴇꜱ ]  --- ( 2 )

ᴀᴅ∥ʙᴄ ᴀɴᴅ ᴀᴀᴄ ᴀꜱ ᴛʀᴀᴠᴇʀꜱᴀʟ,

∴  ∠ᴅᴀᴄ=∠ʙᴄᴀ         [ ᴀʟᴛᴇʀɴᴀᴛᴇ ᴀɴɢʟᴇꜱ ]   --- ( 3 )

ꜰʀᴏᴍ ( 1 ), ( 2 ) ᴀɴᴅ ( 3 )

∠ᴅᴀᴄ=∠ʙᴀᴄ=∠ᴅᴄᴀ=∠ʙᴄᴀ

∴  ∠ᴅᴄᴀ=∠ʙᴄᴀ

ʜᴇɴᴄᴇ, ᴀᴄ ʙɪꜱᴇᴄᴛꜱ ∠ᴄ.