# Polygons in Spherical and Euclidean Geometry Exploration

**Objective:**

Explore the existence of certain types of polygon in Euclidean geometry and spherical geometry. Understand the importance of definitions.

### A. Biangles

In Spherical geometry, two sided polygons (2-gons) exist. They are also called biangles, bi-gons, and lunes.

- Draw an example of a 2-gon on a sphere.
- Why are there no 2-gons in Eucidean geometry? Which axiom is “responsible” for the difference between the two geometries?

### B. Triangles

In Euclidean geometry we can define

- A
**regular triangle**: any 3-gon with congruent sides and angles.

At least two definitions of **equilateral triangle** are possible:

- ET1: a triangle with all 3 sides congruent.
- ET2: a triangle with three 60 degree angles.

Finally, define

- An
**equiangular triangle**: a triangle with three congruent angles.

In Euclidean geometry all four of these definitions describe the same polygons.

- Which of these triangles exist in spherical geometry?
- Of the ones that exist, do they define the same shapes? Or could they be different? Explain.

### C. Squares

In Euclidean geometry we can define a **square** in at least two different ways:

- S1: A 4-gon with congruent sides and congruent angles.
- S2: A 4-gon with congruent sides and all angles measuring 90°.

Compare this to:

- A regular 4-gon: A 4-gon with congruent sides and congruent angles.

In Euclidean geometry these are one and the same thing.

- Which of these shapes exist in spherical geometry?
- Would you say squares exist on the sphere? Why or why not?

### D. Rectangles

In Euclidean geometry we can define a rectangle in several different ways:

- R1: A 4-gon with all interior angles 90°.
- R2: A 4-gon with all interior angles congruent.
- R3: A 4-gon with two pairs of parallel sides and all angles congruent.

- Which of these definitions cannot possibly work on the sphere?
- Is there one that defines 4-gons which are possible to construct on the sphere? What will such a 4-gon look like? What would you call it?

### E. Parallelograms

In Euclidean geometry the following statements can all be used to define a parallelogram. They all describe the exact same family of polygons.

- P1: A quadrilateral with opposite sides parallel and equal in length, and opposite angles equal.
- P2: A quadrilateral with both pairs of opposite sides parallel and equal in length.
- P3: A quadrilateral with opposite sides parallel.
- P4: A quadrilateral with opposite sides congruent (theorem).
- P5: A quadrilateral with opposite angles congruent (theorem).
- P6: A quadrilateral whose diagonals bisect one another (theorem).

- Which definition cannot possibly work on the sphere? Why?
- Of the other definitions, which ones correspond to polygons we can construct on the sphere? Can they be used interchangeably? What would you call these polygons?

**Handin:**
A sheet with answers to all questions.