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To solve for the measures of angles \( x \) and \( y \) in the given triangle configuration, we can apply the following steps.

### Step 1: Analyze the Triangle
The triangle \( ABC \) is an isosceles triangle because sides \( AB \) and \( AC \) are marked as equal. This implies that angles \( ABC \) and \( ACB \) are equal. Let this common angle be denoted as \( \theta \).

Given angles:
- \(\angle BAC = 58^\circ\)
- \(\angle CAD = 16^\circ\)

### Step 2: Calculate \( \theta \)
Since \( ABC \) is isosceles, we have:
\[
\theta + \theta + 58^\circ = 180^\circ \quad \text{(Sum of angles in a triangle)}
\]
\[
2\theta = 180^\circ - 58^\circ = 122^\circ
\]
\[
\theta = \frac{122^\circ}{2} = 61^\circ
\]

Thus, \( \angle ABC = \angle ACB = 61^\circ \).

### Step 3: Calculate Angle \( x \)
Since \( x \) is an external angle of triangle \( ABC \), it is equal to the sum of the non-adjacent interior angles \( \theta \) and \( \angle BAC \). Hence:
\[
x = \theta + \angle BAC = 61^\circ + 58^\circ = 119^\circ
\]

### Step 4: Calculate Angle \( y \)
To find \( y \), we can observe that \( y \) is part of triangle \( ACD \). The angle \( y \) is calculated by subtracting the known angles from \( 180^\circ \) (sum of angles in triangle \( ACD \)):
\[
y = 180^\circ - (\angle CAD + x)
\]
\[
y = 180^\circ - (16^\circ + 119^\circ) = 180^\circ - 135^\circ = 45^\circ
\]

### Final Answers:
- \( x = 119^\circ \)
- \( y = 45^\circ \)

Would you like further details on any step or have any additional questions?

### Related Questions:
1. How do you prove that triangle \( ABC \) is isosceles?
2. What is the significance of external angles in a triangle?
3. How would you solve the problem if the given angles were different?
4. Can you derive angle \( y \) using alternate methods?
5. What are some properties of isosceles triangles that can be useful in other problems?

### Tip:
When working with isosceles triangles, always remember that the base angles are equal, which can simplify calculations.

Learn how to find the measures of angles x and y in a triangle using the properties of isosceles triangles and the sum of angles theorem. This problem involves identifying angles in a specific triangle configuration with given angle measurements.

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