3'5 Inequalities in a Triangle

1
3.5 Inequalities in a Triangle

• Definition of greater than
• Let a and b be real numbers
• agtb (read a is greater than b) if and only if
  there is a positive number p for which a b p.
• Lemmas (Helping Theorems) Theorems used to
  prove other Theorems.
• Lemma 3.5.1 If B is between A and C on AD, the
  AC gt AB then AC gt BC. ( The measure of a line
  segment is greater than the measure of any of its
  parts). Proof p. 160

A
B
C
2
Lemmas Continued

• Lemma 3.5.2 If BD separates ?ABC into two parts
  (?1 and ?2),, then m?ABCgtm?1 and m?ABC gt m?2.
  The measure of an angle is greater than the
  measure of any of its parts. Proof p. 160
• Lemma 3.5.3 If ?3 is an exterior angle of a
  triangle and ?1 and ?2 are the non-adjacent
  interior angles, then m?3 gt m?1 and m?3 gt m?2.
  (The measure of an exterior angle of a triangle
  is greater than the measure of either nonadjacent
  interior angles. Proof p. 160

• A
• D
• 1
• B 2 C
• 1
• 2
  3

3
More Lemmas on Inequality

• A
• B C

• Lemma 3.5.4 In ?ABC, if ?C is a right angle or
  an obtuse angle, then m ?C gt m?A and m ?C gt m?B.
  (If a triangle contains a right or an obtuse
  angle, then the measure of this angle is greater
  than the measure of either of the remaining
  angles)
• Lemma 3.5.5 (Addition Property of Inequality)
  
• If a gt b and cgt d, then a c gt b d.
• Theorem 3.5.6 If one side of a triangle is
  longer than a second side, then the measure of
  the angle opposite the longer side is greater
  than the measure of the angle opposite the
  shorter side. Proof p. 162
• This relations extends so that the longest side
  is opposite the largest angle and the shortest
  side is opposite the smallest angle.

4
More Lemmas on Inequality

• Theorem 3.5.7 (converse of 3.5.6) If the
  measure of one angle of a triangle is greater
  than the measure of a second angle, then the side
  opposite the larger angle is longer than the side
  opposite the smaller angle. Proof Ex 4. p. 163
• Corollary 3.5.8 The perpendicular segment from
  a point to a line is the shortest segment that
  can be drawn from the point to the line. Fig 3.54
  p. 163
• Corollary 3.5.9 (extending 3.5.8 to three
  dimensions) The perpendicular segment from a
  point to a plane is the shortest segment that can
  be drawn from the point to the plane. Fig. 3.55
• Theorem 3.5.10 (Triangle Inequality). The sum
  of the lengths of any two sides of a triangle is
  greater than the length of the third side. (The
  length of any side of a triangle must lie between
  the sum and difference of the lengths of the
  other two sides. Proof p. 164.

5
Triangle Inequality

• The shortest trip from city A to city B is the
  direct route represented by the segment AB. This
  direct route must be less than any indirect
  route, such as that from city A to city C, then
  from city C to city B.

C
B
A