In which of the following cases, a triangle CANNOT be drawn ?

Options

A. An isosceles triangle with the length of each equal side 7.5 cm.
B. BC = 6 cm, `angleB = 90^(@)` and `angleC = 120^(@)`
C. AB = 6 cm, `angleC = 10^(@)` and CA = 5 cm
D. XY = 8 cm, `angleX = 30^(@)` and `angleY = 60^(@)`

Answer

To determine in which of the following cases a triangle cannot be drawn, we need to analyze each option based on the properties of triangles, particularly the angle sum property, which states that the sum of the angles in a triangle must equal 180 degrees.

### Step-by-Step Solution:

1. **Option 1: Isosceles Triangle with Equal Sides**
- Given: Length of each equal side = 7.5 cm.
- An isosceles triangle can be formed with two sides equal and a base. Since we have two sides of equal length, we can easily draw this triangle.
- **Conclusion**: A triangle can be drawn. (This option is valid.)

2. **Option 2: Angles B and C**
- Given: Angle B = 90 degrees, Angle C = 120 degrees.
- According to the angle sum property, the sum of angles in a triangle must equal 180 degrees.
- Calculation: 90 + 120 = 210 degrees, which is greater than 180 degrees.
- **Conclusion**: A triangle cannot be drawn. (This option is invalid.)

3. **Option 3: Side AB and Angle C**
- Given: AB = 6 cm, angle C = 10 degrees, and side CA = 5 cm.
- We can visualize this: angle C is small, and we can draw side CA of 5 cm and then extend AB from point A to 6 cm.
- **Conclusion**: A triangle can be drawn. (This option is valid.)

4. **Option 4: Angles X and Y**
- Given: XY = 8 cm, angle X = 30 degrees, angle Y = 60 degrees.
- Calculation: 30 + 60 = 90 degrees. The third angle will be 180 - 90 = 90 degrees.
- We can draw this triangle as it satisfies the angle sum property.
- **Conclusion**: A triangle can be drawn. (This option is valid.)

### Final Answer:
The option in which a triangle cannot be drawn is **Option 2**.

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