If `triangle ABC ,` is obtuse angled at C , then

Options

A. `AB gt BC`
B. `AB = BC`
C. `AB lt BC`
D. `AC lt BC`

Answer

To solve the problem, we need to analyze the properties of triangle ABC, which is obtuse-angled at angle C. Let's go through the solution step by step.

### Step-by-Step Solution:

1. **Understanding the Triangle**:
We have triangle ABC with angle C being obtuse. An obtuse angle is one that is greater than 90 degrees.

2. **Sum of Angles in a Triangle**:
The sum of the angles in any triangle is always 180 degrees. Therefore, we can write:
\[ \angle A + \angle B + \angle C = 180^\circ \]

3. **Analyzing Angle C**:
Since angle C is obtuse, we know:
\[ \angle C > 90^\circ \]
This implies that the sum of angles A and B must be less than 90 degrees:
\[ \angle A + \angle B = 180^\circ - \angle C < 90^\circ \]

4. **Conclusion about Angles A and B**:
Since the sum of angles A and B is less than 90 degrees, both angles A and B must be acute (less than 90 degrees).

5. **Comparing the Sides**:
In any triangle, the side opposite the largest angle is the longest side. Since angle C is the largest angle (being obtuse), side AB (opposite angle C) must be the longest side. Therefore, we can conclude:
\[ AB > AC \quad \text{and} \quad AB > BC \]

6. **Evaluating the Options**:
- **Option A**: AB is greater than BC. (Correct)
- **Option B**: AB is equal to BC. (Incorrect, as AB is the longest side)
- **Option C**: AB is less than BC. (Incorrect)
- **Option D**: AC is less than BC. (Indeterminate without additional information)

### Final Conclusion:
The correct answer is **Option A: AB is greater than BC**.

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