Q). In the given figure, prove that BC is parallel to ED.

Dear Student,
Here is the solution of your asked query:
In ABC and AED, we have;AD=ABBAC=DAE     {Vertically opposite angles}AE=ACSo, ABCAEDThen by CPCT we have;AED=ACB and ADE=ABCThis means alternate angles are equal which results in BCDE

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As we can see, we need to test the congruency so we will try the tests 
in the figure, we can see that the lines AE and AC are given the same marking, proving that they are equal
same goes with the lines BA and DA and hence we have got two sides
we assume BC andED to be equal be congruency  test SSS
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Hope it helps
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