Question

If ABCD is a quadrilateral in which AB|| CD and AD=BC, prove that.

1. <A=<B
2. AC=BD​

Answer 1

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

Hence Proved