Question

If ad bc are the equal sides of an isosceles trapezium abcd prove that angle a = angle b

Answer 1

Given: ABCD is a trapezium,

To prove: ∠A=∠B

Construction: Extend AB upto E and join E with C.

Proof: AD║CE ( by construction)

and AE║DC

⇒AECD is a parallelogram.

Therefore, AD=CE ( Opposite sides of parallelogram)

But, we are given that AD=BC

⇒BC=CE

⇒∠CEB=∠CBE(  Angles opposite to equal sides are equal)

For, AD║CE and AR being the transversal, we have ∠A+∠CEB=180° (Interior angles on the same side of transversal)

⇒∠A=180°-∠CEB                                                                       (1)

Also, AE is a line, then

∠B+∠CBE=180°                

⇒∠B=180°-∠CBE                                                                       (2)

From (1) and (2),

∠A=180°-∠CEB  ,∠B=180°-∠CBE  

⇒∠A=∠B

Hence proved.

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