Question

in a quadrilateral each pair of opposite angles is equal​

Answer 1

ɢɪᴠᴇɴ: ᴀ ϙᴜᴀᴅʀɪʟᴀᴛᴇʀᴀʟ ᴀʙᴄᴅ ɪɴ ᴡʜɪᴄʜ ᴏᴘᴘᴏsɪᴛᴇ ᴀɴɢʟᴇs ᴀʀᴇ ᴇϙᴜᴀʟ

ɪ.ᴇ., ∠ᴀ = ∠ᴄ ᴀᴅ ∠ʙ = ∠ᴅ  

ᴛᴏ ᴘʀᴏᴠᴇ: ᴀʙᴄᴅ ɪs ᴀ ᴘᴀʀᴀʟʟᴇʟᴏɢʀᴀᴍ ɪ.ᴇ, ᴀʙ ║ ᴅᴄ ᴀɴᴅ ᴀᴅ ║ ʙᴄ.  

ᴘʀᴏᴏғ:

sɪɴᴄᴇ, ᴛʜᴇ sᴜᴍ ᴏғ ᴛʜᴇ ᴀɴɢʟᴇs ᴏғ ϙᴜᴀᴅʀɪʟᴀᴛᴇʀᴀʟ ɪs 360

⇒ ∠ᴀ + ∠ʙ + ∠ᴄ + ∠ᴅ = 360

⇒ ∠ᴀ + ∠ᴅ + ∠ᴀ + ∠ᴅ = 360 [∠ᴀ = ∠ᴄ ᴀɴᴅ ∠ʙ = ∠ᴅ]  

⇒ ⑵∠ᴀ = ⑵∠ᴅ = 360

⇒ ∠ᴀ + ∠ᴅ = 180 [ᴄᴏ-ɪɴᴛᴇʀɪᴏʀ ᴀɴɢʟᴇ]  

⇒ ᴀʙ ║ ᴅᴄ  

sɪᴍɪʟᴀʀʟʏ,  

∠ᴀ + ∠ʙ + ∠ᴄ + ∠ᴅ = 360

⇒ ∠ᴀ + ∠ʙ + ∠ᴀ + ∠ʙ = 360 [∠ᴀ = ∠ᴄ ᴀɴᴅ ∠ʙ = ∠ᴅ]  

⇒ ⑵∠ᴀ + ⑵∠ʙ = 360

⇒ ∠ᴀ + ∠ʙ = 180

[ ∵ᴛʜɪs ɪs sᴜᴍ ᴏғ ɪɴᴛᴇʀɪᴏʀ ᴀɴɢʟᴇs ᴏɴ ᴛʜᴇ sᴀᴍᴇ sɪᴅᴇ ᴏғ ᴛʀᴀɴsᴠᴇʀsᴀʟ ᴀʙ ]  

∴ ᴀᴅ ║ ʙᴄ  

sᴏ, ᴀʙ ║ ᴅᴄ ᴀɴᴅ ᴀᴅ ║ ʙᴄ  

⇒ ᴀʙᴄᴅ ɪs ᴀ ᴘᴀʀᴀʟʟᴇʟᴏɢʀᴀᴍ.