Question

prove that two Triangles are equiangular then their corresponding sides are promotional

Answer 1

ᴄᴏɴsᴛʀᴜᴄᴛɪᴏɴ: ᴀʙᴄ ɪs ᴀ ᴛʀɪᴀɴɢʟᴇ. ᴅᴇ || ʙᴄ ᴀɴᴅ ᴅᴇ ɪɴᴛᴇʀsᴇᴄᴛs ᴀʙ ᴀᴛ ᴅ ᴀɴᴅ ᴀᴄ ᴀᴛ ᴇ.

ᴊᴏɪɴ ʙ ᴛᴏ ᴇ ᴀɴᴅ ᴄ ᴛᴏ ᴅ. ᴅʀᴀᴡ ᴅɴ ⊥ ᴀʙ ᴀɴᴅ ᴇᴍ ⊥ ᴀᴄ.

ᴛᴏ ᴘʀᴏᴠᴇ: ᴀᴅ/ᴅʙ=ᴀᴇ/ᴇᴄ

ᴘʀᴏᴏғ:

ᴀʀ ᴀᴇᴍ=1/2×(ᴀᴅ)×(ᴇᴍ)

sɪᴍɪʟᴀʀʟʏ;

ᴀʀ( ʙᴅᴇ)=1/2×(ᴅʙ)×(ᴇᴍ)

ᴀʀ (ᴀᴅᴇ)=1/2×(ᴀᴇ)×(ᴅɴ)

ᴀʀ( ᴅᴇᴄ)=1/2×(ᴇᴄ)×(ᴅɴ)

ʜᴇɴᴄᴇ;

ᴀʀ ᴀᴅᴇ/ᴀʀ ʙᴅᴇ=1/2×(ᴀᴅ)×(ᴇᴍ)1/2×(ᴅʙ)×(ᴇᴍ)=ᴀᴅ/ᴅʙ

sɪᴍɪʟᴀʀʟʏ;

ᴀʀ ᴀᴅᴇ/ᴀʀ ᴅᴇᴄ=ᴀᴇ/ᴇᴄ

ᴛʀɪᴀɴɢʟᴇs ʙᴅᴇ ᴀɴᴅ ᴅᴇᴄ ᴀʀᴇ ᴏɴ ᴛʜᴇ sᴀᴍᴇ ʙᴀsᴇ, ɪ.ᴇ. ᴅᴇ ᴀɴᴅ ʙᴇᴛᴡᴇᴇɴ sᴀᴍᴇ ᴘᴀʀᴀʟʟᴇʟs, ɪ.ᴇ. ᴅᴇ ᴀɴᴅ ʙᴄ.

ʜᴇɴᴄᴇ, ᴀʀ(ʙᴅᴇ) = ᴀʀ(ᴅᴇᴄ)

ғʀᴏᴍ ᴀʙᴏᴠᴇ ᴇǫᴜᴀᴛɪᴏɴs, ɪᴛ ɪs ᴄʟᴇᴀʀ ᴛʜᴀᴛ;

ᴀᴅ/ᴅʙ=ᴀᴇ/ᴇᴄ ᴘʀᴏᴠᴇᴅ