If `DeltaABC~DeltaDFE,angleA=30^@,angleC=50^(@)`,AB=5 cm,AC=8 cm and DF=7.5 cm. Then, which of the following is true?

Options

A. `DE=12 cm,angleF=50^(@)`
B. `DE=12cm,angleF=100^(@)`
C. `EF=12cm,angleD=100^(@)`
D. `EF=12cm,angleD=30^(@)`

Explanation

To solve the problem, we need to use the properties of similar triangles. Given that triangle ABC is similar to triangle DFE, we can use the corresponding angles and sides to find the required relationships.

### Step-by-Step Solution:

1. **Identify the Angles**:
We know that:
- Angle A = 30°
- Angle C = 50°
Since the sum of angles in a triangle is 180°, we can find Angle B:
\[ \text{Angle B} = 180° - \text{Angle A} - \text{Angle C} = 180° - 30° - 50° = 100° \]

2. **Determine the Corresponding Angles in Triangle DFE**:
Since triangles ABC and DFE are similar, the corresponding angles are equal:
- Angle D = Angle A = 30°
- Angle F = Angle B = 100°
- Angle E = Angle C = 50°

3. **Set Up the Proportions**:
From the similarity of triangles, we can set up the following proportion based on the corresponding sides:
\[ \frac{AB}{DF} = \frac{AC}{DE} = \frac{BC}{EF} \]
We know:
- AB = 5 cm
- AC = 8 cm
- DF = 7.5 cm

4. **Calculate DE**:
Using the proportion:
\[ \frac{AB}{DF} = \frac{AC}{DE} \]
Substituting the known values:
\[ \frac{5}{7.5} = \frac{8}{DE} \]
Cross-multiplying gives:
\[ 5 \cdot DE = 8 \cdot 7.5 \]
\[ 5 \cdot DE = 60 \]
Dividing both sides by 5:
\[ DE = \frac{60}{5} = 12 \text{ cm} \]

5. **Conclusion**:
Now we have found DE = 12 cm. We can compare the sides and angles to determine the truth of the statements provided in the options.