Final answer:
Angle BAD is congruent to angle CAD because AD, being perpendicular to BC, forms two mirrored isosceles triangles where angles BAD and CAD are at the base of each respective triangle.
Step-by-step explanation:
The reason that justifies the statement that angle BAD is congruent to angle CAD when segment AD is perpendicular to segment BC is due to the definition of a perpendicular bisector and the properties of an isosceles triangle. When a line is perpendicular to a segment at its midpoint, it not only bisects the segment but also creates two congruent angles adjacent to that segment. Thus, if AD is perpendicular and bisects BC, then triangles ABD and ACD are mirror images, making ∠BAD ≅ ∠CAD.