# Congruent Triangles

Angle-Side-Side (ASS)a. Are two triangles congruent if an angle, an adjacent side, and the opposite side of one triangle are congruent to an angle, an , adjacent side, and the opposite side of the other! Look at plane, spheres, and hyperbolic planes.b. Show that ASS holds for right triangles on the plane {where the Angle in Angle-Side-Side is right).This result is often called the Right-Leg-Hypotenuse Theorem (RLH), which can be expressed in the following way:RLH: On the plane, if the leg and hypotenuse of one righttriangle are congruent to the leg and hypotenuse of anotherright triangle, then the triangles are congruent.What happens on a sphere and a hyperbolic plane?SuggestionsSuppose you have two triangles with the above congruencies. We will call them ASS triangles. We would like to see if, in fact, the triangles are congruent. We can line up the angle and the first side, and we know the length of the second side (BC or B’Cj, but we don’t know where the second and third sides will meet.Here, the circle that has as its radius the second side of the triangle intersects the ray that goes from A along the angle a to B twice. So ASS doesn’t work for all triangles on the plane or spheres or hyperbolic plane6. Try this for yourself on these surfaces to see what happens. Can you make ASS work for an appropriately restricted class of triangles? On a sphere, also look at triangles with multiple right angles, and, again, define “small triangles” as necessary. Your definition of “small triangle”

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