Let's break down each part of your question:

1. Cutting a Square-Based Cake into Two Congruent Parts

To divide a square-based cake into two congruent parts, the cuts must go through the center of the square and result in two identical halves. Here are the possible ways:

• Horizontal cut: Slicing the cake parallel to the base (dividing the cake into a top and bottom half).
• Vertical cut: Cutting the cake perpendicular to the base, either along the width or the length (creating left and right or front and back halves).
• Diagonal cuts: Slicing the cake from one corner of the square base to the opposite corner (dividing it into two triangular prisms).

Therefore, for a square-based cake, there are 4 distinct ways to cut it into two congruent parts:

1. Horizontal cut
2. Vertical cut along the width
3. Vertical cut along the length
4. Diagonal cut from corner to corner

2. Using Eight Straight Lines to Form a Square and Four Congruent Equilateral Triangles

To make this configuration with eight straight lines, follow this concept:

• Square: Use four straight lines of equal length to form the sides of the square.
• Equilateral Triangles: Use the remaining four straight lines to form triangles inside the square. This can be done by drawing two diagonals across the square, and then using the other two lines to form additional equilateral triangles inside the square.

I can generate a drawing to illustrate this structure. Would you like me to proceed with that?

Further Exploration Questions:

1. How many ways can you cut a circular cake into two congruent parts?
2. What are the different types of symmetry present in a square-based cake?
3. How many ways can you divide a rectangular cake into congruent parts?
4. How can you modify the configuration of straight lines to form different shapes within the square?
5. How can cutting techniques vary if the cake has multiple layers or toppings?

Tip: When trying to visualize symmetrical cuts or geometric shapes, always consider how symmetry aligns with both planes and angles.