Question:1. Suppose that a triangle and a rectangle lie in a plane. What is the greatest number of points at which they can intersect?2. Suppose that a circle and a square lie in a plane. What is the greatest number of points at which they can intersect?3. Suppose two distinct triangles lie in a plane. What is the greatest number of points at which they can intersect?4. Suppose that a circle and a triangle lie in a plane. What is the least number of points at which they can intersect?5. Suppose two distinct squares lie in a plane. What is the least number of points at which they can intersect?

These are five questions and five answers:

I attache a pdf file with drawings for each question showing the answer and below the explanation for each drawing.

1. Suppose that a triangle and a rectangle lie in a plane. What is the greatest number of points at which they can intersect?

Answer: 6.

Because one line of the triangle can intersect maximum two lines of the rectangle, which makes two intersection points.

So, the maximum number of possible intersections is when you arrange the triangles so that its three lines intersect three different lines each.

See the attached picture.

2. Suppose that a circle and a square lie in a plane. What is the greatest number of points at which they can intersect?

Answer: 8.

The attached figure shows a circle and a square with 8 intersection points.

That is the maximum number of points at which a circle and a square in a plane can intersect: each line of the square intersect two different points of the circle.

3. Suppose two distinct triangles lie in a plane. What is the greatest number of points at which they can intersect?

Answer: 6

See the image attached.

Each line of a triangle intersect one different line of the other triangle in two different points.

4. Suppose that a circle and a triangle lie in a plane. What is the least number of points at which they can intersect?

Answer: 0

You can draw two figures that do not intersect each other. See the picture attached.

5. Suppose two distinct squares lie in a plane. What is the least number of points at which they can intersect?

Answer: 0

As you can see the figure attached you can draw two different squares which to not intersect each other.