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Lecture 18: Side-Angle-Side
18.1 Congruence
Notation: Given 4 ABC, we will write ∠A for ∠BAC, ∠B for ∠ABC, and ∠C for ∠BCA if the meaning is clear from the context.
Definition In a protractor geometry, we write 4 ABC ' 4DEF if
AB ' DE, BC ' EF , CA ' F D,
and ∠A ' ∠D, ∠B ' ∠E, ∠C ' ∠F.
Definition In a protractor geometry, if 4 ABC ' 4DEF , 4 ACB ' 4DEF , 4 BAC ' 4 DEF , 4 BCA ' 4DEF , 4 CAB ' 4DEF , or 4 CBA ' 4DEF , we say 4 ABC and 4 DEF are congruent
Example In the Taxicab Plane, let A = (0, 0), B = (− 1 , 1), C = (1, 1), D = (5, 0), E = (5, 2), and F = (7, 0). Then AB = 2 = DE,
AC = 2 = DF,
m(∠A) = 90 = m(∠D),
m(∠B) = 45 = m(∠E),
and m(∠C) = 45 = m(∠F ),
and so AB ' DE, AC ' DF , ∠A ' ∠D, ∠B ' ∠E, and ∠C ' ∠F. However,
BC = 2 6 = 4 = EF.
Thus BC and EF are not congruent, and so 4 ABC and 4 DEF are not congruent.
18.2 Side-angle-side
Definition A protractor geometry satisfies the Side-Angle-Side Axiom (SAS) if, given 4 ABC and 4 DEF , AB ' DE, ∠B ' ∠E, and BC ' EF imply 4 ABC ' 4DEF.
Definition We call a protractor geometry satisfying the Side-Angle-Side Axiom a neutral geometry, also called an absolute geometry.
Lecture 18: Side-Angle-Side 18-
Example We will show that the Euclidean Plane is a neutral geometry. First recall the Law of Cosines: Given any triangle 4 ABC in the Euclidean Plane,
AC^2 = AB^2 + BC^2 − 2(AB)(BC) cos(mE (∠B)).
Note that, in particular, the measure of any angle of a triangle in the Euclidean Plane is determined by the lengths of the sides of the triangle. Hence given 4 ABC and 4 DEF with AB ' DE, ∠B ' E, and BC ' EF , we need show only that AC ' DF to conclude that 4 ABC ' 4DEF. Now
AC^2 = AB^2 + BC^2 − 2(AB)(BC) cos(mE (∠B)) = DE^2 + EF 2 − 2(DE)(EF ) cos(mE (∠E)) = DF 2 ,
so AC = DF. Thus AC ' DF.
Example The Poincar´e Plane is a neutral geometry. We will omit the proof, which is more easily done with the help of an axiom about isometries of the plane which is equivalent to the Side-Angle-Side Axiom.
18.3 Isosceles triangles
Definition In a protractor geometry, we say a triangle with two congruent sides is isosceles. We say a triangle which is not isosceles is scalene. It 4 ABC is isosceles with AB ' BC, then we call ∠A and ∠B the base angles of 4 ABC. We say a triangle with all three sides congruent is equilateral.
Pons Asinorum In a neutral geometry, the base angles of an isosceles triangle are congruent.
Proof Consider a triangle 4 ABC with AB ' BC. Then AB ' CB, ∠ABC ' ∠CBA, and BC ' BA. Hence, by Side-Angle-Side, 4 ABC ' 4CBA. In particular, ∠BAC ' ∠BCA.
