AAS Exploration Hyperbolic Geometry
- Author:
- Dr. Jack L. Jackson II
- Topic:
- Congruence, Geometry, Triangles, Trigonometry
AAS Triangle Condition Exploration in Hyperbolic Geometry
Manipulate the sliders and/or input boxes to specify the measures of two angles and the length of the side opposite the first angle.
Are there conditions that the measurements must have in order for the triangle to exist?
Slide the Step slider slowly, one step at a time, to see the construction unfold.
Unlike Euclidean Geometry, there is no fixed angle sum in Hyperbolic Geometry. Therefore, we cannot simply compute the remaining angle size and reduce to the ASA case, like we can in Euclidean Geometry. The point C? shows up on the appropriate ray. We have two ways to directly explore the AAS condition from there. The line segment forming a side of the triangle must go from C to A. However, this forms the angle alpha. Manipulate C ? until alpha has the measure of Angle C given by the slider. Alternately, we can form a ray from C ? that has the appropriate Angle C. However, this ray must go though A to complete the triangle. Move C ? along the ray to make the green ray go through A and alpha equal the measure of Angle C. How many places does this happen?
In Hyperbolic Geometry, if we are given two triangles with two pair of congruent corresponding angles, and the pair of corresponding sides opposite one of these angles are also congruent (AAS Condition), then do the two triangles have to be congruent?