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The problem in this image involves analyzing and calculating various properties of a triangle \( ABC \) with angles and side lengths specified. Let's work through each section systematically.

### Information Given:
- \( \angle BAC = 60^\circ \)
- \( \angle ABC = 45^\circ \)
- \( \angle BCA = 75^\circ \)
- \( AB = 20 \sqrt{3} = 34.6 \, \text{cm} \)

#### Questions:
1. **Angle Calculations**:
   - Calculate the values of angles \( \angle BAC \), \( \angle BAD \), and \( \angle CAD \) using the given information.
2. **Side Lengths**:
   - \( BD = \frac{1}{2} \times AB \) and \( AD = \frac{1}{2} \times \sqrt{3} \times AB \)
3. **Pythagorean Relation**:
   - Calculate \( AC \) using the formula \( AC^2 = AD^2 + DC^2 \).
4. **Area of Triangle**:
   - Use the area formula \( \frac{1}{2} \times \text{base} \times \text{height} \) for \( \triangle ABC \).

I’ll go through each of these calculations based on the steps presented in the worksheet:

#### Solution:
1. **Angle Calculations**:
   - Since \( \triangle ABC \) is divided by \( AD \), and it specifies \( \angle BAC = 60^\circ \), we should verify each angle step-by-step based on given information and typical trigonometric relationships.

2. **Length Calculations**:
   - \( BD = \frac{1}{2} \times AB = \frac{1}{2} \times 34.6 = 17.3 \, \text{cm} \)
   - \( AD = \frac{1}{2} \times \sqrt{3} \times AB = \frac{1}{2} \times \sqrt{3} \times 34.6 \approx 30 \, \text{cm} \)

3. **Calculate \( AC \)**:
   - Using \( AC^2 = AD^2 + DC^2 \), substitute \( AD \) and \( DC \) based on values obtained above to find \( AC \).

4. **Area Calculation**:
   - Calculate the area \( \text{Area of } \triangle ABC = \frac{1}{2} \times \text{base} \times \text{height} \).

Would you like further steps or calculations for any part? 

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### Related Questions

1. What are the exact trigonometric values of angles commonly used in such problems?
2. How does the sine rule apply to non-right triangles like \( \triangle ABC \)?
3. How would we determine the area if only side lengths were given (e.g., using Heron's formula)?
4. How could the Pythagorean theorem help in calculating unknown sides?
5. What is the significance of the \( \frac{1}{2} \sqrt{3} \) factor in calculating \( AD \) from \( AB \)?

#### Tip
When working with triangles involving angles like 45°, 60°, and 30°, special trigonometric values simplify calculations greatly.

Given a triangle ABC with angle BAC = 60°, angle ABC = 45°, angle BCA = 75°, and side length AB = 20√3 = 34.6 cm, find the values for angle BAC, side lengths BD, AD, AC, and the area of triangle ABC.

Explore the solution to finding the angles, side lengths, and area of a triangle ABC with given angles of 60°, 45°, and 75°, and side length AB = 20√3 = 34.6 cm. This solution uses trigonometry, the Pythagorean theorem, and triangle properties, ideal for students in Grades 10-12.

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