Euclid's triangles
What's my line?
Transformation Time!
On the hyperbolic plane
Potent Postulates
100
This formula relates the hypotenuse of a right triangle to the other two sides and is attributed to an ancient cult leader.
What is the Pythagorean theorem?
100
The name for two lines which never intersect.
What is parallel?
100
An isometry is defined by the fact that is preserves this.
What is distance?
100
This quadrilateral has two equal sides perpendicular to its base; its other two angles are acute in hyperbolic geometry.
What is a Saccheri quadrilateral?
100
Any line segment can be extended indefinitely into a straight line.
What is Postulate 2?
200
This triangle has two congruent sides.
What is an isosceles triangle?
200
This property of a set requires the line segment connecting any two points in the set to lie inside the set.
What is a convex set?
200
The type of isometry which transforms the left foot into the right foot below.
What is a glide reflection?
200
These lines are all parallel to d on the Poincare disc model below.
What are p, q, and n?
200
Postulate 8.
What is Plane Separation?
300
In a triangle, this angle is greater than either of the nonadjacent interior angles.
What is the exterior angle?
300
This theorem states that given a point P on line L, P divides L into two non-empty disjoint sets and does not hold in spherical geometry.
What is line separation?
300
The name for a point which is mapped to itself under an isometry.
What is a fixed point?
300
This triangle congruence relation does not hold in Euclidean geometry, but can be used to show that all similar triangles are congruent in hyperbolic geometry.
What is angle-angle-angle?
300
On attempting to prove this postulate, Wolfgang Bolyai said "Do not waste an hour's time on that problem. It does not lead to any result; instead it will come to poison all your life."
What is the parallel postulate (or Postulate 5)?
400
In triangle ABC, this relation says AB + BC > AC.
What is the triangle inequality?
400
In Euclidean geometry, the distance between two parallel lines is always _____.
What is the same?
400
In the dihedral group D10 generated by a rotation R of 72 degrees and a reflection F, this is the value of x that makes the statement true:
R2F = FRx
What is x=3?
400
This figure in hyperbolic geometry is described by four points with a [ symbol and has no equivalent in Euclidean geometry.
What is a biangle?
400
Postulate 4
What is all right angles are congruent?
500
These are all of the congruence relations in Euclidean geometry.
What are SAS, SSS, ASA, and AAS?
500
The failure of this property means that being parallel does not constitute an equivalence relation.
What is reflexivity?
500
This palace in Granada was constructed during the Moorish rule of Spain and boasts numerous examples of highly symmetric tile patterns prevalent in Islamic art and design.
What is the Alhambra?
500
This triangle has three asymptotic parallels as sides.
What is an ideal triangle?
500
Circular Continuity principle
What is Postulate 11?