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.