For which of the following conditions the construction of a triangle is NOT possible?

Options

A. If two sides and one angle is given.
B. If two sides and included angle between them is given.
C. If three sides are given.
D. If two angles and side between them is given.

Answer

To determine the conditions under which the construction of a triangle is not possible, we will analyze each of the given options step by step.

### Step-by-Step Solution:

1. **Option A: Two sides and one angle (not included)**
- Given two sides (let's say AB and AC) and an angle (∠A) that is not between them.
- In this case, we can draw the two sides AB and AC from point A, but we cannot determine where point C will lie because the angle is not included between the two sides.
- Therefore, we cannot complete the triangle construction as we do not know the position of point C.
- **Conclusion**: Triangle construction is not possible with this condition.

2. **Option B: Two sides and an included angle**
- Given two sides (let's say AB and AC) and the included angle (∠A).
- Here, we can construct the triangle by drawing side AB and then using the angle ∠A to determine the position of point C.
- **Conclusion**: Triangle construction is possible with this condition.

3. **Option C: Three sides**
- Given three sides (let's say lengths a, b, and c).
- We can construct the triangle by drawing one side (say AB), and then using the lengths of the other two sides to find point C.
- **Conclusion**: Triangle construction is possible with this condition.

4. **Option D: Two angles and the side between them**
- Given two angles (let's say ∠A and ∠B) and the side (AB) between them.
- We can construct the triangle by drawing the side AB and then using the angles to find points A and B.
- **Conclusion**: Triangle construction is possible with this condition.

### Final Answer:
The condition under which the construction of a triangle is NOT possible is **Option A: Two sides and one angle (not included)**.

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