If a translation maps ∠E onto ∠A, which of the following statements is true?

triangle CAB, point E is on segment AC between points A and C and point D is on segment BC between points B and C, creating segment ED

 ∠B ≅ ∠C
 ∠B ≅ ∠D
 segment CE ≅ segment CA
 segment CD ≅ segment CB

Given the problem, we are working with a translation that maps \(\angle E\) onto \(\angle A\). Let’s break this down.

You are provided with a triangle \(CAB\), where point \(E\) lies on segment \(AC\) and point \(D\) lies on segment \(BC\). These points create segment \(ED\) within the triangle.

**Translation Properties:**
A translation in geometry means that every point of a figure is moved the same distance in the same direction. Since translations preserve the size and shape of figures, corresponding angles and segments remain congruent.

From the problem:
- Since the translation maps \(\angle E\) onto \(\angle A\), these angles must be congruent: 
  \[
  \angle E \cong \angle A
  \]
- The following choices need to be evaluated:

1. **\(\angle B \cong \angle C\):**  
   There is no information suggesting that \(\angle B\) and \(\angle C\) are congruent.

2. **\(\angle B \cong \angle D\):**  
   Since \(D\) lies on segment \(BC\), there is no indication that \(\angle B\) and \(\angle D\) are congruent either.

3. **Segment \(CE \cong\) Segment \(CA\):**  
   This could potentially be true based on the placement of point \(E\) and the idea of congruence under translation, but the statement doesn't clearly indicate this.

4. **Segment \(CD \cong\) Segment \(CB\):**  
   Given that point \(D\) lies between \(B\) and \(C\), and segment \(CB\) includes both, this is also a likely valid statement since the translation could have mapped the points accordingly.

Thus, **the correct statement is likely**:
\[
\boxed{\text{Segment } CD \cong \text{Segment } CB}
\]

This is based on the fact that point \(D\) lies on \(BC\) and the congruence from the translation would likely apply.

---

Would you like further details or have any questions?

### Related Questions:
1. What are the properties of translations in geometry?
2. How do congruent angles behave under translations?
3. Can translations affect the orientation of a triangle?
4. What other types of transformations preserve congruence?
5. How does the position of point \(D\) affect the segments of triangle \(CAB\)?

**Tip:** Always remember that translations preserve distances and angles, which means corresponding sides and angles remain congruent.

If a translation maps ∠E onto ∠A, which of the following statements is true? In triangle CAB, point E is on segment AC between points A and C, and point D is on segment BC between points B and C, creating segment ED. The following options are provided: ∠B ≅ ∠C, ∠B ≅ ∠D, segment CE ≅ segment CA, or segment CD ≅ segment CB.

Explore how translations in geometry affect angles and segments within a triangle. This problem involves a triangle CAB and analyzes which statements about angles and segments are true under a translation that maps ∠E onto ∠A. Suitable for students in Grades 8-10.

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