You solve it by looking at the figure to determine what corresponding sides and angles you have, and what their relationships are to each other. A side that is shared between two triangles is a congruent side, as it is always congruent to itself.
a. ∠BDA ≅ ∠BDC = 90° (a perpendicular line creates two congruent right angles); BD ≅ BD (as discussed above); ∠ABD ≅ ∠CBD (marked on the diagram). This means you have two angles and the side between them that are congruent.
ΔADB ≅ ΔCDB by the ASA postulate. (angle-side-angle: two angles and the side between them)
b. DB ≅ AC (marked); ∠DBC ≅ ∠ACB (marked); BC ≅ CB (as discussed above). This means you have two sides and the angle between them that are congruent.
ΔDBC ≅ ΔACB by the SAS postulate. (side-angle-side: two sides and the angle between them)
c. Alternate interior angles are congruent, so ∠A ≅ ∠D and ∠B ≅ ∠E. Vertical angles are congruent, so ∠ACB ≅ ∠DCE. However, sides BC and DC are not corresponding sides, so there is not enough information here to say anything except that triangles ACB and DCE are similar.
d. ∠BCA ≅ ∠ADB (marked); ∠CAB ≅ ∠DBA (marked); AB ≅ BA (see above). This means you have two angles and a side not between them that are congruent.
ΔABC ≅ ΔBAD by the AAS postulate. (angle-angle-side: two angles and the side not between them.)
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In general, if you have three congruent corresponding parts, including at least one side, there is a postulate that will say the triangles are congruent. The exception is that SSA will only work if the angle given is the largest angle (which may be a right angle). In the case of a right angle, this postulate is referred to as the HL postulate (hypotenuse, leg).