In circle O above, `AB=3 and CD=2`. Which of the following statements must be true?
I. `angleABC and angleBCD` have equal measures
II. `overline(AB)||overline(CD)`
III. `2(AE)=3(CE)`

Options

A. I and II
B. I and III
C. I only
D. III only

Answer

To solve the problem, we need to analyze the statements given in the context of the circle and the segments AB and CD.

### Step-by-Step Solution:

1. **Understanding the Circle and Segments**:
- We have a circle with center O.
- The segments AB and CD are given lengths: AB = 3 and CD = 2.

2. **Analyzing Statement I**:
- Statement I claims that angles ∠ABC and ∠BCD have equal measures.
- These angles intercept the same arc BD. By the property of angles in a circle that intercept the same arc, we can conclude that ∠ABC = ∠BCD.
- Therefore, Statement I is **True**.

3. **Analyzing Statement II**:
- Statement II claims that segments AB and CD are parallel (AB || CD).
- There is no information provided in the problem that indicates AB and CD are parallel. Thus, we cannot assume this to be true.
- Therefore, Statement II is **False**.

4. **Analyzing Statement III**:
- Statement III claims that 2(AE) = 3(CE).
- To analyze this, we can use the property of similar triangles.
- By the angle-angle similarity criterion, triangles ABE and CDE are similar because they share angles and have equal angles from the intercepted arcs.
- From the similarity of triangles, we have:
\[ \frac{AE}{CE} = \frac{AB}{CD} \]
Substituting the known values:
\[ \frac{AE}{CE} = \frac{3}{2} \]
- Cross-multiplying gives us:
\[ 2(AE) = 3(CE) \]
- Therefore, Statement III is **True**.

5. **Conclusion**:
- The only statements that must be true are I and III.
- Thus, the correct answer is that statements I and III are true.

### Final Answer:
- Statements I and III are true.