Download Theorem list for these sections.

Absolute geometry
Other theorems of absolute geometry
Material for this section references College Geometry: A Discovery Approach, 2/e, David C. Kay, Addison Wesley, 2001. In particular, see sections 3.5 and 3.6, pp
166 - 181.
These sections have familiar theorems; be sure to look at the proofs in the text. Here’s the theorem
list:
Scalene inequality
If one side of a triangle has greater length than another side, then the angle opposite
the longer side has the greater angle measure, and, conversely, the side opposite an angle
having the greater measure is the longer side.
[Kay, p 166]
• You knew this one.
• Lets you order sides and angles
AC < AB < BC
if and only if
m∠ B < m∠ C < m∠ A
• Proof depends on isosceles triangle theorem, and exterior angle inequality, which in turn depend
on SAS postulate.
Triangle inequality
If A, B, and C are any three distinct points, then AB + BC ≥ AC, with equality only
when the points are collinear, and A − B − C.
quad [Kay, p 167]
• You knew this one too.
• Geometric interpretation is that the sum of the lengths of any two legs of a triangle must be
greater than the length of the third leg.
• Proof depends on the scalene inequality (above), which depends on isosceles triangle theorem,
and exterior angle inequality, which depend on SAS.
Hinge theorem (SAS inequality)
If in 4ABC and 4XY Z we have AB = XY , AC = XZ, but m∠ A > m∠ X, then
BC > Y Z, and, conversely, if BC > Y Z, then m∠ A > m∠ X.
[Kay, p. 169]
• Way to compare between two triangles - side opposite wider angle is longer side, and vice versa.
• Extends scalene inequality (which only allows comparisons within one triangle, instead of between
two triangles).
• I don’t think we’ve worked with this explicitly, but it’s pretty self evident (you’d probably assume
it without even thinking about it).
• Depends directly on SAS.
AAS congruence criterion
If under some correspondence of their vertices, two angles and a side opposite in one
triangle are congruent to the corresponding two angles and side of a second triangle, then
the triangles are congruent.
[Kay, p 174]
• We’re used to an immediate result of AAS from ASA - if two angles line up, we subtract the
third from 180, those match two, and the side falls between two angles. This can’t be used here,
because we don’t have angle sum equal to 180 guaranteed.
• But it can still be proven (proof by contradiction, plus a construction - very similar to the proof
of ASA) that AAS congruence gives congruent triangles. Doesn’t rely on angle sum of 180 at all.
• Depends on ASA congruence theorem, which depends on SAS postulate.
SSA theorem
If under some correspondence of their vertices, two triangles have two pairs of corresponding
sides and a pair of corresponding angles congruent, and if the triangles are not congruent
under this correspondence, then the remaining pair of angles not included by the congruent
sides are supplementary angles.
[Kay, p 175]
∠ C and ∠ D are supplementary.
m∠ C + m∠ D = 180◦
• SSA correspondence cannot be used to prove triangles congruent.
• However, there are only two possible cases. You’ve seen this before - it’s the swinging leg effect.
One triangle has an acute angle, the other obtuse.
• The SSA theorem says there’s a relationship; if the triangles aren’t congruent (one of the possibilities), then then non-included angles must be supplementary (the other possibility).