Question

use the diagram to complete the statements. segment ad must be congruent to segment segment bc must be congruent to segment

Explanation:

Step1: Identify congruent triangles

Since \(AB = AB\) (common side), \(\angle ABD=\angle ABE = 90^{\circ}\), and \(BD = BE\) (given from the diagram's perpendicular bisector - like markings), by the Hypotenuse - Leg (HL) congruence criterion for right - triangles (\(\triangle ABD\) and \(\triangle ABE\)), \(\triangle ABD\cong\triangle ABE\). So \(AD = AE\).

Step2: Identify another pair of congruent triangles

Since \(AB = AB\) (common side), \(\angle ABC=\angle ABE = 90^{\circ}\), and \(BD = BE\) (given from the diagram's markings), \(\triangle ABC\cong\triangle ABE\) (by HL if we consider right - triangles or by Side - Angle - Side if we use the fact that the segments are equal and the right angles). Also, from \(\triangle ABD\cong\triangle ABE\) and \(\triangle ABC\cong\triangle ABE\), we can say that \(\triangle ABD\cong\triangle ACE\) (by transitivity of congruence in a more complex geometric sense considering the overall figure). Another way: If we assume that the lines are constructed such that for the triangles involving \(BC\) and \(DE\), since the triangles \(\triangle ABC\) and \(\triangle ADE\) (after establishing other congruences) have corresponding parts equal. In a more basic sense, looking at the symmetry of the figure (assuming the perpendicular and equal - segment markings), \(BC = DE\).

Answer:

Segment \(AD\) must be congruent to segment \(AE\). Segment \(BC\) must be congruent to segment \(DE\).