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To solve for the angles \( \angle BCD \) and \( \angle DCA \) in this triangle, we can apply trigonometric rules, specifically the Law of Cosines or Law of Sines, depending on the information provided.

### Given Information:
- \( u = v = 22 \)
- \( w = 29 \)
- \( \angle CAD = 30^\circ \)

#### Steps to Solve:
1. **Find \( \angle BCD \)**:
   - We can use the Law of Cosines in triangle \( \triangle BCD \) since we know the lengths \( u = 22 \), \( v = 22 \), and the angle \( \angle CAD = 30^\circ \).
   - Using the Law of Cosines:
     \[
     \cos(\angle BCD) = \frac{u^2 + v^2 - w^2}{2 \cdot u \cdot v}
     \]
   - Substituting the values:
     \[
     \cos(\angle BCD) = \frac{22^2 + 22^2 - 29^2}{2 \cdot 22 \cdot 22}
     \]
   - Simplifying this will give us \( \angle BCD \).

2. **Find \( \angle DCA \)**:
   - Once \( \angle BCD \) is determined, we can find \( \angle DCA \) because the sum of angles in \( \triangle BCD \) is \( 180^\circ \).
   - Calculate \( \angle DCA = 180^\circ - \angle BCD - \angle CAD \).

Let's go through these calculations.

### Calculations

#### Step 1: Calculate \( \angle BCD \)
\[
\cos(\angle BCD) = \frac{22^2 + 22^2 - 29^2}{2 \cdot 22 \cdot 22}
\]
\[
= \frac{484 + 484 - 841}{968} = \frac{127}{968} \approx 0.1312
\]
\[
\angle BCD \approx \cos^{-1}(0.1312) \approx 82.4^\circ
\]

#### Step 2: Calculate \( \angle DCA \)
\[
\angle DCA = 180^\circ - 82.4^\circ - 30^\circ = 67.6^\circ
\]

### Final Answers
- \( \angle BCD \approx 82.4^\circ \)
- \( \angle DCA \approx 67.6^\circ \)

Would you like additional details on any of these steps?

### Related Questions:
1. How would you use the Law of Sines in this problem?
2. Why is the Law of Cosines useful in triangles without a right angle?
3. How does knowing one angle help determine the other angles?
4. What would happen if \( u \) and \( v \) were different lengths?
5. Can these methods be used on any triangle?

### Tip:
In triangles with known two side lengths and the included angle, start with the Law of Cosines for more straightforward angle calculation.

For the triangle shown, find the following. (Assume u = v = 22 and w = 29. Round your answers to one decimal place.)
(a) ∠BCD
(b) ∠DCA

Solve for the angles ∠BCD and ∠DCA in a triangle with given sides u = 22, v = 22, and w = 29 using the Law of Cosines. Detailed solution with trigonometric calculations and angle summation. Suitable for Grades 10-12.

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