# Hyperbolic Geometry II with NonEuclid Exploration

NonEuclid is a simulation that allows you to draw lines and circles in the Hyperbolic Plane.

We will investigate some of the theorems and facts from Euclidean Geometry. Your job is to determine which of the statements are also theorems and facts in Hyperbolic Geometry.

## Contents

## Angles and triangles

*What is a hyperbolic angle?:*
Hyperbolic Angles are formed by the intersection of Hyperbolic rays analogous to the formation of angles in Euclidean Geometry. The measure of a Hyperbolic angle, BAC is defined to be the measure of the Euclidean angle, B'AC', formed by the Euclidean tangent lines, AB' and AC'.

*Measuring an angle in noneuclid:*
You can measure a Hyperbolic angle by selecting the "Measure Angle" option from the "Measurement" Menu.

Note that you can also choose “measure triangle" from the measurements menu. This will compute all lengths of sides, angles, and angle sum for you.

A **(hyperbolic) triangle** is a closed figure formed by three line segments.

**Question 1:** Construct a triangle. Measure the triangle. Move the vertices around. What are the largest and smallest angle sums you can find?

**Question 2:** True/False: The following is a list of theorems about Triangles in Euclidean Geometry. Which (if any) are theorems in Hyperbolic Geometry?

(A) The sum of the angles of a Triangle is 180 degrees.

(B) The longest side of a Triangle is opposite the greatest angle.

(C) All three altitudes of a Triangle intersect in a single point. (Hint: To construct an altitude of a triangle, use the "Draw Perpendicular" command from the "Constructions" menu. Click the mouse on any two vertices to define the base. Then click on the third vertex to draw the altitude.)

(D) In a triangle, the sum of any two sides is always greater than the length of the third side.

**Question 3:**In a Euclidean Triangle, the product "base times height" is the same regardless of which side is chosen as the base. For example, in triangle ABC, (AB) x (the height to C) = (BC) x (the height to A). Give example of the construction you did, and the measurements you found. Give a sketch

## Equilateral Triangles

An **Equilateral Triangle** is a triangle that has three sides of equal length.

Use the “draw segments of a specific length” command from the constructions menu to create an equilateral triangle.

**Question 4:**Determine if the following statements from Euclidean Geometry are valid in Hyperbolic Geometry (Indicate True/False):
(A) It is possible to construct an Equilateral Triangle.

(B) An Equilateral Triangle is also Equiangular (all three angles have equal measure).

(C) Each angle of an Equilateral Triangle measures 60 degrees.

## Rhombus

A **Rhombus** is a quadrilateral in which all four sides have equal length.

**Question 5:** It is possible to construct a Rhombus. Construct one for yourself. Sketch what this polygon looks like in Hyperbolic Geometry.

**Question 6:**The following is a list of theorems about Rhombi in Euclidean Geometry. Which (if any) are theorems in Hyperbolic Geometry? Indicate if the statements are true or false.

(A) The opposite angles of a Rhombus are congruent.

(B) The diagonals of a Rhombus bisect each other.

(C) The diagonals of a Rhombus are perpendicular.

(D) The diagonals of a Rhombus bisect the Rhombus's angles.

## Rectangles and Squares

A **rectangle** is a quadrilateral where all four of its angles are right angles (i.e. measure 90 degrees).
A ({define|square}} is a quadrilateral which has four right angles and parallel sides.

**Question 7:** The following is a list of theorems about Rhombi in Euclidean Geometry. Which (if any) are theorems in Hyperbolic Geometry? Indicate if the statements are true or false.
(A) It is possible to construct a Rectangle.
(B) It is possible to construct a Square.

**Question 8:** In Euclidean geometry there's a theorem that says that in a square all sides are congruent and all angles are congruent. If we take this as the definition of a square, do they exist?

## Parallelograms

A **parallelogram** is a quadrilateral with two sets of parallel sides.

**Question 9:** It is possible to construct a Parallelogram. Construct one for yourself. Sketch what this polygon looks like in Hyperbolic Geometry.

**Question 10:** In Euclidean geometry there's a result which says that the opposite sides of a parallelogram are of equal length, and the opposite angles of a parallelogram are congruent.

Is this also true in hyperbolic geometry? Show why or why not (draw examples).

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

Are opposite angles congruent? Include your measurements for your parallelogram.