In `triangleABC, AB gt BC` and BC `gt` AC. Then, the smallest angle of `triangle`ABC is ..........

Options

A. `angle A`
B. `angle C`
C. `angle B`
D. `angle A` or `angle C`

Answer

To solve the problem, we need to determine which angle in triangle ABC is the smallest based on the given conditions about the sides of the triangle.

### Step-by-step Solution:

1. **Identify the Given Information**:
We have a triangle ABC where:
- \( AB > BC \)
- \( BC > AC \)

2. **Understand the Relationship Between Sides and Angles**:
In any triangle, the side lengths are directly related to the angles opposite them. Specifically:
- The larger the side, the larger the angle opposite that side.
- Conversely, the smaller the side, the smaller the angle opposite that side.

3. **Assign the Sides to Angles**:
Based on the given inequalities:
- Since \( AB > BC \), the angle opposite \( AB \) (which is angle \( C \)) must be greater than the angle opposite \( BC \) (which is angle \( A \)).
- Since \( BC > AC \), the angle opposite \( BC \) (which is angle \( A \)) must be greater than the angle opposite \( AC \) (which is angle \( B \)).

4. **Establish the Order of Angles**:
From the relationships established:
- \( \angle C > \angle A \) (because \( AB > BC \))
- \( \angle A > \angle B \) (because \( BC > AC \))

Therefore, we can conclude:
- \( \angle C > \angle A > \angle B \)

5. **Identify the Smallest Angle**:
From the established order of angles, it is clear that:
- \( \angle B \) is the smallest angle in triangle ABC.

### Final Answer:
The smallest angle of triangle ABC is \( \angle B \).

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