The ratio of the angles in `Delta ABC` is 2 : 3 : 4. Which of the following triangles is similar to `Delta ABC` ?

Options

A. `DeltaDEF` has angle in the ratio 4 : 3 : 2
B. `DeltaPQR` has angle in the ratio 1 : 2 : 3
C. `DeltaLMN` has angle in the ratio 1 : 1 : 1
D. `DeltaMRQ` has angle in the ratio 3 : 2 : 4

Answer

To solve the problem, we need to find which triangles are similar to triangle ABC based on the given ratio of angles, which is 2:3:4.

### Step-by-Step Solution:

1. **Understanding the Ratio of Angles**:
The angles of triangle ABC are in the ratio of 2:3:4. Let's denote the angles as:
- Angle A = 2x
- Angle B = 3x
- Angle C = 4x

2. **Finding the Value of x**:
The sum of angles in any triangle is 180 degrees. Therefore, we can set up the equation:
\[ 2x + 3x + 4x = 180 \]
Simplifying this gives:
\[ 9x = 180 \]
Dividing both sides by 9, we find:
\[ x = 20 \]

3. **Calculating the Angles**:
Now we can find the actual angles of triangle ABC:
- Angle A = 2x = 2(20) = 40 degrees
- Angle B = 3x = 3(20) = 60 degrees
- Angle C = 4x = 4(20) = 80 degrees

4. **Identifying Similar Triangles**:
We need to check which of the given triangles have angles that are in the same ratio as 40:60:80 or equivalently 2:3:4.

- **Option A**: Triangle DEF has angles in the ratio 4:3:2. This does not match.
- **Option B**: Triangle TQR has angles in the ratio 1:2:3. This does not match.
- **Option C**: Triangle LMN has angles in the ratio 1:1:1. This does not match.
- **Option D**: Triangle MRQ has angles in the ratio 3:2:4. This does not match.

5. **Conclusion**:
Based on the analysis, the triangles that are similar to triangle ABC are those that have angles in the same ratio of 2:3:4. Therefore, the correct options are:
- Triangle DEF (4:3:2) is not similar.
- Triangle TQR (1:2:3) is not similar.
- Triangle LMN (1:1:1) is not similar.
- Triangle MRQ (3:2:4) is not similar.

Thus, the answer is that none of the given options are similar to triangle ABC based on the angle ratios provided.