Download Inequality Theorems If we extend side B C of ΔABC to locate a point

Inequality Theorems
If we extend side B C of ΔABC to locate a point D
so that B-C-D, then ∠ACD is said to be an exterior
angle of the triangle; it is supplementary to the
adjacent interior angle ∠ACB. The other
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interior angles, ∠ABC and ∠BAC, are the
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opposite interior angles to ∠ACD.
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In Euclidean geometry, any exterior angle to a
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triangle has a measure equal to the sum of the
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measures of its opposite interior angles. But this
fact cannot be proved from the axioms we currently
have at our disposal. To see this, we exhibit a
model for the plane geometry we have built so far
in which this exterior angle property fails to hold.
In the Poincaré model of geometry, space consists
of all points interior to some (Euclidean) circle Γ
with center O. The lines in this geometry are
either diameters of the circle or arcs of circles that
intersect the boundary of Γ at right angles (see
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Figure 3.29, p. 153). While we delay for later the
description of how distance is defined in this model,
the normal notion of betweenness of points on a
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line is used here. Further, angle measure is the
same as in Euclidean geometry, noting that the
angle between an arc and a line, or between two
arcs, is the angle made between the Euclidean
tangent lines to those arcs at the vertex of the
angle. You should check that all our axioms for
plane geometry hold in the Poincaré model (for
now, you must accept on faith that the distance
axioms will also hold). However, it is not had to
show that the exterior angle property fails in this
geometry (see Figure 3.30, p. 154).
Consider another model, given by spherical
geometry: here, space consists of all points on a
sphere in Euclidean space and lines are great
circles on that sphere. Distance is the normal
arclength of circles as measured in Euclidean
geometry, and angles between great circles have
the same measures as the angles made by the
tangent lines to these curves at the vertex. While
axiom I-1 fails in this geometry (in fact, some pairs
of points lie on infinitely many distinct lines!),
axiom I-5 and the distance and angle axioms do
hold. However, the exterior angle property does
not (see Figure 3.31, p. 155).
In both models, while the exterior angle property
fails, it fails in the same way, namely, in every
case, the exterior angle is never smaller than the
sum of the measures of the opposite interior angles.
This follows from a theorem we can prove:
Theorem [Exterior Angle Inequality] An
exterior angle of a triangle has measure greater
than either of the opposite interior angles. //
Corollary The sum of the measures of any two
angles of a triangle is less than 180. //
Corollary A triangle can have at most one angle
that is not acute. //
Corollary The base angles of an isosceles triangle
must be acute. //
Theorem [Saccheri-Legendre] The angle sum of
any triangle is never greater than 180. //
Theorem [Scalene Inequality] The angle
opposite the longer side in a triangle has greater
measure; conversely, the side opposite the greater
angle in a triangle is longer. //
Corollary The hypotenuse is the longest side in
any right triangle. //
Corollary The side opposite an obtuse angle in a
triangle is the longest side. //
Theorem [Triangle Inequality] Given any three
points A, B, C, then AB + BC ≥ AC, with equality if
and only if A-B-C. //
Corollary [Median Inequality] Any median
A M of the triangle ΔABC is shorter than the
average of the lengths of the sides A B and A C
between which it lies. //
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Theorem [SAS Inequality] If triangles ΔABC
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and ΔXYZ satisfy AB = XY and AC = XZ, then
m∠A < m∠X if and only if BC < YZ. //
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Theorem [AAS implies congruence] If ΔABC
and ΔXYZ are triangles for which ∠A ≅ ∠X ,
∠B ≅ ∠Y , and B C ≅ Y Z , then ΔABC ≅ ΔXYZ . //
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And while ASS (!) is not a hypothesis that implies
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congruence, we can assert the following:
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Theorem If ΔABC and ΔXYZ are noncongruent
triangles for which ∠A ≅ ∠X , A B ≅ XY , and
B C ≅ Y Z , then ∠C and ∠Z are supplementary. //
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Corollary If ΔABC and ΔXYZ are acute-angled
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triangles
for
which
∠A
≅ ∠X , A B ≅ XY , and
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B C ≅ Y Z , then ΔABC ≅ ΔXYZ . //
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Corollary [SsA implies congruence] If ΔABC
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and ΔXYZ
are
triangles
for which ∠A ≅ ∠X ,
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A B ≅ XY , B C ≅ Y Z , and BC ≥ AB, then ΔABC ≅
ΔXYZ . //
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Corollary [HL Theorem] If two right triangles
have congruent hypotenuses and one pair of
congruent legs, then the triangles are congruent. //
Corollary [HA Theorem] If two right triangles
have congruent hypotenuses and one pair of
congruent acute angles, then the triangles are
congruent. //
Corollary [LA Theorem] If two right triangles
have one pair of congruent acute angles and one
pair of corresponding congruent legs, then the
triangles are congruent. //