Question

Line l is a perpendicular bisector of line segment A C. It intersects line segment A C at point B. Line l also contains points D and E. Line segments A D, D C, C E, and E A are shown.

Use the diagram to complete the statements.

Segment AD must be congruent to segment
.

Segment BC must be congruent to segment
.

Answer 1

In the given scenario, since line \( l \) is the perpendicular bisector of line segment \( AC \) and intersects it at point \( B \), we can deduce the following:

1. **For segment \( AD \)**: Since line \( l \) bisects \( AC \) at point \( B \), it means that \( AB \) is equal to \( BC \). Additionally, the fact that line \( l \) is perpendicular to \( AC \) implies that triangle \( ABD \) is congruent to triangle \( CBE \) (by the Hypotenuse-Leg theorem or other congruent triangle theorems). Therefore, we can conclude that segment \( AD \) must be congruent to segment \( CE \).

2. **For segment \( BC \)**: Since \( B \) is the midpoint of \( AC \), the segment \( BC \) is equal to \( AB \) as \( l \) bisects \( AC \). We can therefore conclude that segment \( BC \) must be congruent to segment \( AB \).

Based on this reasoning, the completed statements would be:
- Segment \( AD \) must be congruent to segment \( CE \).
- Segment \( BC \) must be congruent to segment \( AB \).