The triangle in the image has a given angle of $111^\circ$ and is part of an isosceles triangle since two of its sides are marked with equal lengths. Let's solve for the angle $k^\circ$ using the fact that the sum of the internal angles of a triangle is always $180^\circ$.

Step-by-step solution:

1. Sum of angles in a triangle: The sum of the angles in any triangle is $180^\circ$.

2. Given angle: One of the angles is $111^\circ$.

3. Isosceles property: Since the triangle is isosceles, the two other angles are equal. Let these angles be $k^\circ$.

4. Equation for angle sum: $k + k + 111^\circ = 180^\circ$ Simplifying: $2k + 111^\circ = 180^\circ$ Subtract $111^\circ$ from both sides: $2k = 69^\circ$ Divide by 2: $k = 34.5^\circ$

Thus, the angle $k = 34.5^\circ$.

Let me know if you'd like more details or have any further questions!

Here are 5 related questions for you to consider:

1. What is the total sum of the angles in a quadrilateral?
2. How do you find the angles of a right triangle?
3. What properties do equilateral triangles have in terms of their angles?
4. How do you use the sine rule in non-right-angled triangles?
5. What is the difference between isosceles and scalene triangles?

Tip: In any isosceles triangle, the two base angles are always equal. This is a key property that can help you solve many geometry problems quickly!