Question

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Answer 1

ᴀsᴀ(ᴀɴɢʟᴇ sɪᴅᴇ ᴀɴɢʟᴇ):

ᴛᴡᴏ ᴛʀɪᴀɴɢʟᴇs ᴀʀᴇ ᴄᴏɴɢʀᴜᴇɴᴛ ɪғ ᴛᴡᴏ ᴀɴɢʟᴇs ᴀɴᴅ ᴛʜᴇ ɪɴᴄʟᴜᴅᴇᴅ sɪᴅᴇ ᴏғ ᴏɴᴇ ᴛʀɪᴀɴɢʟᴇ ᴀʀᴇ ᴇϙᴜᴀʟ ᴛᴏ ᴛᴡᴏ ᴀɴɢʟᴇs & ᴛʜᴇ ɪɴᴄʟᴜᴅᴇᴅ sɪᴅᴇ ᴏғ ᴛʜᴇ  ᴏᴛʜᴇʀ ᴛʀɪᴀɴɢʟᴇ.

ᴀʟᴛᴇʀɴᴀᴛᴇ ᴀɴɢʟᴇs:

ᴡʜᴇɴ ᴛᴡᴏ ʟɪɴᴇs ᴀʀᴇ ᴄʀᴏssᴇᴅ ʙʏ ᴀɴᴏᴛʜᴇʀ ʟɪɴᴇ ᴛʜᴇ ᴘᴀɪʀ ᴏғ ᴀɴɢʟᴇs ᴏɴ ᴏᴘᴘᴏsɪᴛᴇ sɪᴅᴇs ᴏғ ᴛʜᴇ ᴛʀᴀɴsᴠᴇʀsᴀʟ ɪs ᴄᴀʟʟᴇᴅ ᴀʟᴛᴇʀɴᴀᴛᴇ ᴀɴɢʟᴇs.

 

ᴜsᴇ ᴛʜᴇ ᴘʀᴏᴘᴇʀᴛɪᴇs ᴏғ ᴀ ᴛʀᴀɴsᴠᴇʀsᴀʟ ɪɴᴛᴇʀsᴇᴄᴛɪɴɢ ᴛᴡᴏ ᴘᴀʀᴀʟʟᴇʟ ʟɪɴᴇs ɪ.ᴇ, ɪɴᴛᴇʀɪᴏʀ ᴀɴɢʟᴇs ᴀʀᴇ ᴇϙᴜᴀʟ ᴛᴏ ᴇᴀᴄʜ ᴏᴛʜᴇʀ, ᴛᴏ sʜᴏᴡ ᴄᴏɴɢʀᴜᴇɴᴄᴇ ᴏғ ɢɪᴠᴇɴ ᴛʀɪᴀɴɢʟᴇs.

 

ɢɪᴠᴇɴ,

ʟ || ᴍ ᴀɴᴅ ᴘ || ϙ

ᴛᴏ ᴘʀᴏᴠᴇ,

Δᴀʙᴄ ≅ Δᴄᴅᴀ

ᴘʀᴏᴏғ,

ɪɴ Δᴀʙᴄ ᴀɴᴅ Δᴄᴅᴀ,

∠ʙᴄᴀ = ∠ᴅᴀᴄ (ᴀʟᴛᴇʀɴᴀᴛᴇ ɪɴᴛᴇʀɪᴏʀ ᴀɴɢʟᴇs ᴀs ᴘ||ϙ)

ᴀᴄ = ᴄᴀ (ᴄᴏᴍᴍᴏɴ)

∠ʙᴀᴄ = ∠ᴅᴄᴀ (ᴀʟᴛᴇʀɴᴀᴛᴇ ɪɴᴛᴇʀɪᴏʀ ᴀɴɢʟᴇs ᴀs ʟ ||ᴍ)

ʜᴇɴᴄᴇ, Δᴀʙᴄ ≅ Δᴄᴅᴀ (ʙʏ ᴀsᴀ ᴄᴏɴɢʀᴜᴇɴᴄᴇ ʀᴜʟᴇ.)

Answer 2

Answer:

ASA ( Angle side angle )-

Two triangles are congruent if two angles and the included side of the one triangle are equal to two angles and the included side of the other Triangle.

Alternate Angle -

When two lines are crossed by another line the pair of angles on opposite side of the transversal is called alternate angle.

Use the property of transversal:

Intersecting two parallel lines i.e. , interior angles are equal to each other, to show congruence of Given Triangle.

Given,

L|| M and P || Q

To prove ,

∆ABC congruence to ∆CDA

Prove,

In ∆ABC and ∆CDA

⊾BCA = ⊾DCA

Ac = Ca ( Same or Common side)

⊾BCA = ⊾DCA ( Alternate Interior Angles L || M)

Thus,

∆ABC congruence to ∆CDA .