Question

ᴘʀᴏᴠᴇ- ɪɴ ᴀ ᴘᴀʀᴀʟʟᴇʟᴏɢʀᴀᴍ, ᴏᴘᴘᴏsɪᴛᴇ ᴀɴɢʟᴇs ᴀʀᴇ ᴇϙᴜᴀʟ.

ᴛʜᴏʀᴇ ᴛʜᴀɴᴋs ᴅᴇ ᴅɪʏᴀ ᴋʀᴏ sʙ......xᴅ​

Answer 1

Step-by-step explanation:

To prove :-

• Opposite angles of parallelogram are equal.

$\underline{\huge \sf {Prove\: :}}$

Let, Parallelogram be ABCD and BD be diagonal of parallelogram.

We know,

Opposite sides of parallelogram are equal and parallel.

So, AD = BC and AD || BC

And, CD = AB and CD || AB.

In ∆ABD and ∆CDB :

AB = CD [Opposite sides of parallelogram are equal]

BD = DB [Common]

AD = Bc [Opposite sides of parallelogram are equal]

By SSS congruency

∆ABD ≌ ∆CDB

By CPCT

∠A = ∠C

∠CDB = ∠ABD -----(i)

∠ADB = ∠CBD ------(ii)

Add equation (i) and (ii)

➝ ∠CDB + ∠ADB = ∠ABD + ∠CBD

➝ ∠D = ∠B

∠A, ∠B, ∠C and ∠D are angles of parallelogram.

∠A is opposite to ∠C.

∠B is opposite to ∠D.

So,

∠A = ∠C

∠D = ∠B

Hence, Proved!!

Opposite angles of parallelogram are equal.

Answer 2

☆Answer☆

Given: A parallelogram ABCD in which AB||CD and AD||BC.

To prove: Opposite angles are equal  i.e. ∠A = ∠C and ∠B = ∠D 

Construction: Draw diagonal AC. 

Proof:

In ∆ABC and ∆CDA:

∠BAC = ∠DCA [Alternate angles] 

∠BCA = ∠DAC [Alternate angles] 

AC = AC [Common] 

∴ ∆ABC ≅ ∆CDA [By ASA] 

⇒ ∠B = ∠D [By cpctc]  And, ∠BAD = ∠DCB

i.e., ∠A = ∠C 

Similarly, we can prove that ∠B = ∠D

Hence, Opposite angles of tge parallelogram are equal.

PROVED.