Lesson 33: Inequalities and Geometric Inequalities

1
Lesson 33 Inequalities and Geometric Inequalities

• Nov. 16, 2009

2
Homework

• Will unequals multiplying/dividing unequals
  produce inequality in the same order or produce
  equality? Can you give any examples to support
  your conclusions?
• Geometry, P206-207, 6-13

3
Do Now What is inequality and what are the basic
inequality postulates?

• Definition of Greater Than
• Trichotomy Postulate of Inequality
• Transitive Postulate of Inequality
• Addition Postulate of Inequality
• Subtraction Postulate of Inequality
• Multiplication Postulate of Inequality
• Division Postulate of Inequality
• Substitution Postulate of Inequality

4
1. Definition of Greater Than

• Suppose a and b are real numbers. agtb if and only
  if there is a positive real number c such that a
  bc. Also, blta is equivalent to agtb.
• Examples real numbers, segments, angles

5
Example1 How do we use basic inequality
postulates to prove geometric inequality
relationships?

• Prove that the measure of an exterior angle of a
  triangle is greater than the measure of either
  remote interior angles. (Theorem 6-1)

6
2. Trichotomy Postulate

• Suppose a and b are real numbers. Either agtb,
  altb, or ab.

7
3. Transitive Postulate of Inequality

• Suppose a, b and c are real numbers. If agtb and
  bgtc, then agtc.

8
Example 2 How do we use basic inequality
postulates to prove geometric inequality
relationship?

• Given In ?ABC, ABgtAC and M is the midpoint
  of segment AC.
• Prove ABgtAM

9
4. Addition Postulate of Inequality

• Suppose a, b, c and d are real numbers.
• If agtb, then acgtbc
• If agtb and cgtd, then acgtbd

10
5. Subtraction Postulate of Inequality

• Suppose a, b, and c are real numbers. If agtb,
  then a-cgtb-c.
• Q What happens when unequals are subtracted from
  unequals?

11
Example 3 How do we use basic inequality
postulates to prove geometric inequality
relationship?

• Given m?ABCltm?EFG. D and H are points inside
  ?ABC and ?EFG respectively. ?DBC??HFC
• Prove ?ABDlt?EFH

12
6. Multiplication Postulate of Inequality

• Suppose a, b and c are real numbers.
• If agtb and cgt0, then acgtbc.
• If agtb and clt0, then acltbc.
• Q Will unequals multiplying/dividing unequals
  produce inequality in the same order or produce
  equality? Can you give any examples to support
  your conclusions? (Homework)

13
7. Division Postulate of Inequality

• Suppose a, b and c are real numbers.
• If agtb and cgt0, then a/cgtb/c.
• If agtb and clt0, then a/cltb/c.
• Q Will unequals multiplying/dividing unequals
  produce inequality in the same order or produce
  equality? Can you give any examples to support
  your conclusions? (Homework)

14
8. Substitution Postulate of Inequality

• Suppose a, b and c are real numbers. If agtb and
  ac, then cgtb.

15
Example 4 How do we use basic inequality
postulates to prove geometric inequality
relationship?

• Given ABltCD. X and Y are the midpoints of
  segments AB and CD respectively.
• Prove AXltCY