Question

ABCD is a parallelogram, AD is produced to E so that DE =DC and EC produced in F. Prove that BF = BC

Answer 1

Hello... ☺

ɢɪᴠᴇɴ: ᴀʙᴄᴅ ɪs ᴀ ᴘᴀʀᴀʟʟᴇʟᴏɢʀᴀᴍ.
ᴀᴅ ɪs ᴘʀᴏᴅᴜᴄᴇᴅ ᴛᴏ ᴇ sᴜᴄʜ ᴛʜᴀᴛ ᴅᴇ = ᴅᴄ.
ᴇᴄ ɪs ᴘʀᴏᴅᴜᴄᴇᴅ ᴛᴏ ɪɴᴛᴇʀsᴇᴄᴛ ᴀʙ ᴘʀᴏᴅᴜᴄᴇᴅ ɪɴ ғ.

ᴛᴏ ᴘʀᴏᴠᴇ: ʙғ = ʙᴄ

ᴘʀᴏᴏғ:

ɪɴ Δᴅᴄᴇ,

ᴅᴇ = ᴅᴄ  (ɢɪᴠᴇɴ)

∴ ∠ᴅᴄᴇ = ∠ᴅᴇᴄ  ...(1) 
 (ɪɴ ᴀ ᴛʀɪᴀɴɢʟᴇ, ᴇǫᴜᴀʟ sɪᴅᴇs ʜᴀᴠᴇ ᴇǫᴜᴀʟ ᴀɴɢʟᴇs ᴏᴘᴘᴏsɪᴛᴇ ᴛᴏ ᴛʜᴇᴍ)

ᴀʙ || ᴄᴅ
 (ᴏᴘᴘᴏsɪᴛᴇ sɪᴅᴇs ᴏғ  ᴛʜᴇ ᴘᴀʀᴀʟʟᴇʟᴏɢʀᴀᴍ ᴀʀᴇ ᴘᴀʀᴀʟʟᴇʟ)

∴ ᴀғ || ᴄᴅ 
(ᴀʙ ʟɪᴇs ᴏɴ ᴀғ)

ᴀғ || ᴄᴅ ᴀɴᴅ ᴇғ ɪs ᴛʜᴇ ᴛʀᴀɴsᴠᴇʀsᴀʟ,

∴ ∠ᴅᴄᴇ = ∠ʙғᴄ  ... (2) 
(ᴘᴀɪʀ ᴏғ ᴄᴏʀʀᴇsᴘᴏɴᴅɪɴɢ ᴀɴɢʟᴇs)

ғʀᴏᴍ (1) ᴀɴᴅ (2)

∠ᴅᴇᴄ = ∠ʙғᴄ

ɪɴ Δᴀғᴇ,

∠ᴀғᴇ = ∠ᴀᴇғ (∠ᴅᴇᴄ = ∠ʙғᴄ)

∴ ᴀᴇ = ᴀғ
 (ɪɴ ᴀ ᴛʀɪᴀɴɢʟᴇ, ᴇǫᴜᴀʟ ᴀɴɢʟᴇs ʜᴀᴠᴇ ᴇǫᴜᴀʟ sɪᴅᴇs ᴏᴘᴘᴏsɪᴛᴇ ᴛᴏ ᴛʜᴇᴍ)

⇒ ᴀᴅ + ᴅᴇ = ᴀʙ + ʙғ

⇒ ʙᴄ + ᴀʙ = ᴀʙ + ʙғ 
(ᴀᴅ = ʙᴄ, ᴅᴇ = ᴄᴅ ᴀɴᴅ
ᴄᴅ = ᴀʙ ⇒ ᴀʙ = ᴅᴇ)

⇒ ʙᴄ = ʙғ

thank you ♥

Answer 2

ANSWER↓

R.E.F image

Given : ABCD is A parallelogram

To Prove : BF=BC

Proof : In △DCE,DE=DC (given)

∴∠DCE=∠DEC...(1)

(Equal sides have equal is opposite to them)

since,

AB∥CD,∠DCE=∠BFC...(2) (pair of corresponding ∠S)

Form (1) and (2)

∠DEC=∠BFC

In △AEF,∠AEF=∠AFE

∴AF=AE,

⇒AB+BF=AD+DE

⇒BF=AD [∵AB=CD=DE]

⇒BF=BC [∵AD=BC] Hence proved.