Segment CD is the perpendicular bisector of AB at E. Which pair of segments does not have to be congruent?

 a
DE,CE
 b
AE,BE
 c
AD,BD
 d
AC,BC


Given that segment **CD** is the perpendicular bisector of **AB** at point **E**, let's analyze each pair of segments:

1. **DE, CE**: Since **DE** and **CE** are both parts of the perpendicular bisector, there's no requirement for them to be congruent, as **D** and **C** could be different distances from **E**.  
   **These segments do not have to be congruent.** (This could be the correct answer.)

2. **AE, BE**: Since **CD** is a perpendicular bisector, **AE** and **BE** must be congruent, as **E** is the midpoint of **AB**.  
   **These segments must be congruent.**

3. **AD, BD**: Since **CD** is the perpendicular bisector and if **A** and **B** lie symmetrically with respect to **D**, **AD** and **BD** must be congruent.  
   **These segments must be congruent.**

4. **AC, BC**: If **C** is on the perpendicular bisector, then **AC** and **BC** must be congruent due to symmetry.  
   **These segments must be congruent.**

Therefore, the pair of segments that does not have to be congruent is:
**a. DE, CE**

Would you like more details or have any questions?

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Here are 5 related questions for further exploration:

1. What are the properties of perpendicular bisectors in a triangle?
2. How do you construct a perpendicular bisector for a given line segment?
3. What is the definition of congruent line segments?
4. How can you prove that two line segments are congruent using geometry?
5. What is the relationship between perpendicular bisectors and the circumcenter of a triangle?

**Tip:** In a triangle, the perpendicular bisectors of the sides intersect at the circumcenter, which is equidistant from the vertices.

Segment CD is the perpendicular bisector of AB at E. Which pair of segments does not have to be congruent? 

a) DE, CE
b) AE, BE
c) AD, BD
d) AC, BC

Learn how to identify which pair of segments does not have to be congruent when segment CD is the perpendicular bisector of AB at point E. This problem explores key geometric concepts such as the properties of perpendicular bisectors and congruent segments, suitable for Grades 7-9.

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