Let's analyze the problem step-by-step:

Given:

• The problem shows a regular hexagon with an equilateral triangle attached to one of its sides.
• We need to calculate angles $k$ and $y$.

Step 1: Understanding the Geometry of a Regular Hexagon

1. A regular hexagon has 6 equal sides, and all internal angles are $120^\circ$.
2. Since the hexagon is regular, each internal angle is $120^\circ$.

Part (a): Finding Angle $k$

1. The equilateral triangle has internal angles of $60^\circ$.
2. Angle $k$ is formed by the subtraction of the triangle's angle from the hexagon's angle. $k = 120^\circ - 60^\circ = 60^\circ$

So, $k = 60^\circ$.

Part (b): Finding Angle $y$

1. Angle $y$ is part of the straight line formed by the adjacent sides of the hexagon. The sum of angles on a straight line is $180^\circ$.
2. Since angle $k$ is $60^\circ$, angle $y$ will be: $y = 180^\circ - k = 180^\circ - 60^\circ = 120^\circ$

Therefore, $y = 120^\circ$.

Final Answers:

• $k = 60^\circ$
• $y = 120^\circ$

Would you like a more detailed explanation or have any further questions?

Here are 5 related questions:

1. What is the sum of the interior angles of a hexagon?
2. How do you calculate the interior angle of any regular polygon?
3. What is the significance of equilateral triangles in geometry problems?
4. Can we generalize this approach to find angles in other polygons with attached equilateral triangles?
5. How would this problem change if the hexagon was irregular?

Tip: Always remember that regular polygons have equal side lengths and equal interior angles, which makes calculations easier!