1. It is given that line segments AB and DC are parallel. Also, as the line segments AB and CD are congruent they are equal in length.
The parallel line segments AB and DC are intersected by the transversal BD. Angles `/_` ABD` ` and `/_` CDB are alternate interior angles and therefore equal.
In triangles `Delta BAD` and `Delta DCB` , sides AB and CD are equal in length and side BD is common. The angle `/_` ABD formed by AB and BD and the angle `/_` CDB formed by CD and DB are equal. Using SAS it can be proved that the triangles BAD and DCB are congruent.
2. It is given that the triangles ABD and BDC are equilateral. To prove that triangles ABD and BDC are congruent use SSS. As the side BD is common to the two equilateral triangles all the sides in the figure are equal in length. This gives AB = BD, BD = DC and DA = CB. The two triangles ABD and BDC are congruent.
3. It is given that point G is the midpoint of FC and AD. The segments AG and GD are equal and the segments FG and GC are equal. Now take the triangles GAF and GDC, GA = GD, FG = CG. Also, the angle FGA and CGD is a vertical pair and equal. By SAS the two triangles GAF and GDC are congruent.