To solve the problem, I'll first analyze the provided geometric shapes and the equations for angles. It appears that the task involves finding the values of the variables $x$, $y$, and $z$.

For Diagram (e):

The figure is likely a quadrilateral, and the angles inside a quadrilateral sum up to $360^\circ$. The given angles are:

1. $x + 30^\circ$,
2. $4y$,
3. $2x - 60^\circ$,
4. $z$.

So, the equation becomes: $(x + 30) + 4y + (2x - 60) + z = 360$ Simplify: $3x + 4y + z - 30 = 360$ 3x + 4y + z = 390 \tag{1}

Additional constraints may depend on further information about the sides or relationships of angles, which I'll analyze below.

For Diagram (f):

This figure appears to be a triangle, as it contains three angles:

1. $z$,
2. $y$,
3. $60^\circ$.

The angles in a triangle sum up to $180^\circ$. Thus: $z + y + 60 = 180$ Simplify: z + y = 120 \tag{2}

Solving the System of Equations:

Using equation (1) from diagram (e) and equation (2) from diagram (f), we can solve for the variables. Let me calculate the solutions explicitly.

from sympy import symbols, Eq, solve

Define variables

x, y, z = symbols('x y z')

Equations

eq1 = Eq(3x + 4y + z, 390) # From quadrilateral (e) eq2 = Eq(z + y, 120) # From triangle (f)

Solve the system of equations

solution = solve([eq1, eq2], (x, y, z)) solution{x: z - 30, y: 120 - z}{x: z - 30, y: 120 - z}successThe solution to the system of equations is expressed as follows:

$x = z - 30$ $y = 120 - z$

This means the values of $x$, $y$, and $z$ depend on the specific value of $z$, which might be determined by additional geometric constraints or conditions not visible in the problem.

If you'd like to proceed further or clarify any conditions, let me know!

Questions for Expansion:

1. How do the angles relate to side lengths in the given quadrilateral or triangle?
2. Are there any additional geometric constraints not explicitly stated (e.g., isosceles or parallel sides)?
3. Would you like a step-by-step guide on solving the equations for a fixed value of $z$?
4. Are there other geometric relationships in these figures that can help refine the values of $x$, $y$, and $z$?
5. Should we validate if the obtained relationships hold true for any specific scenario?

Tip:

Always verify the total sum of interior angles in polygons using their general formula: $(n-2) \times 180^\circ$, where $n$ is the number of sides.