ABCD is a trapezium in which AB∥CD and AD=BC (see the given figure). Show that ΔABC≅ΔBAD.
Extend AB such that it intersects CE at point E where CE∥AD.
Since, AB∥CD and CE∥AD, therefore, AECD is a parallelogram.
Since, ABCD is a trapezium, therefore,
⇒∠ABC+∠CBE=180∘ [Linear Pair]
Since, AECD is a parallelogram, therefore, ∠BEC=∠ADC=θ ...(iii)
From (ii) and (iii)
Therefore, ∠DAE=180∘−θ...(iv) [Angles on same side of transversal are supplementary]
Now, in ΔABC and ΔBAD.
∠DAB=∠ABC=180∘−θ [From (i) and (iv)]
ΔABC≅ΔBAD by SAS rule.