Math, asked by kumarsumant51, 4 months ago

PI
4)if ABCD is a quadrilateral in which AB||DC and AD=BC,prove that
A=B​

Answers

Answered by mahek77777
24

Here, AB∥CD [Given]

⇒ and AD∥EC [By construction]

∴ AECD is a parallelogram.

⇒ AD = EC [Opposite sides of

parallelogram are equal]

⇒ But AD = EC [Given]

∴ EC = BC

∴ ∠CBE = ∠CEB ---- ( 1 )

⇒ ∠B + ∠CBE = 180° [Linear pair] ---- ( 2 )

⇒ AD∥EC [By construction]

⇒ and transeversal AE intersects them

∴ ∠A + ∠CEB = 180° ---- ( 3 )

[Sum of adjacent angles of parallelogram is supplementary ]

⇒ ∠B + ∠CEB = 180° / [From ( 2 ) and ( 3 )]

⇒ But ∠CBE = ∠CEB [From ( 1 )]

∴ ∠A=∠B [Proved] -- ( 4 )

⇒ ∵ AB∥CD

⇒ ∠A + ∠D = 180°

[Sum Supplementary angles of parallelogram is 180°]

⇒ and ∠B + ∠C = 180°

∴ ∠A + ∠D = ∠B + ∠C

⇒ But ∠A = ∠B [From ( 4 )]

∴ ∠C=∠D

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